SPIRAL  AND  WORM 
GEARING 


SPIRAL  AND  WORM 
GEARING 


A  TREATISE  ON  THE  PRINCIPLES,  DI- 
MENSIONS, CALCULATION  AND  .DESIGN 
OF  SPIRAL  AND  WORM  GEARING,  TO- 
GETHER WITH  CHAPTERS  ON  THE 
METHODS  OF  CUTTING  THE  TEETH 
IN  THESE  TYPES  OF  GEARS 


COMPILED  AND  EDITED 
BY 

ERIK   OBERG,   A.S.M.E. 

ASSOCIATE  EDITOR  or  MACHINERY 
EDITOR  OF  MACHINERY'S  HANDBOOK 

AUTHOR  OF  "  HANDBOOK  OF  SMALL  TOOLS,"  "  SHOP  ARITHMETIC  FOR  THE  MACHINIST," 

"  SOLUTION  OF  TRIANGLES,"  "  STRENGTH  OF  MATERIALS," 

"ELEMENTARY  ALGEBRA,"  ETC. 


FIRST  EDITION 

FIFTH    PRINTING 


NEW  YORK 

THE    INDUSTRIAL    PRESS 

LONDON:    THE   MACHINERY  PUBLISHING  CO.,  LTD. 
IQ20 


COPYRIGHT,  1914 

BY 

THE  INDUSTRIAL  PRESS 
NEW  YORK 


PREFACE 


THE  manner  in  which  MACHINERY'S  book,  <cSpur  and  Bevel 
Gearing,"  has  been  received  by  the  mechanical  world  has 
prompted  the  compilation  and  publication  of  a  companion  book 
on  "  Spiral  and  Worm  Gearing."  This  subject  has  often  been 
presented  in  so  theoretical  a  manner  that  many  have  assumed 
it  to  be  very  difficult  to  master.  It  is  possible,  however,  to 
present  the  principles  of  design  and  calculation  of  spiral  and 
worm  gearing  in  such  a  way  that  they  can  be  readily  under- 
stood without  resorting  to  a  highly  theoretical  treatment;  and 
in  preparing  this  book,  the  first  consideration  on  the  part  of 
the  editor  has  therefore  been  to  treat  the  subject  in  such  a  way 
as  to  meet  the  practical  requirements  of  the  machine-building 
trade. 

As  a  result,  in  this  book,  as  well  as  in  the  companion  book, 
"Spur  and  Bevel  Gearing,"  mere  theory  and  academic  discus- 
sions have  been  avoided.  The  rules,  formulas  and  instruc- 
tions given  are  illustrated  with  engravings  whenever  necessary, 
and  numerous  examples  are  given  to  show  their  application  to 
problems  met  with  in  machine  design.  Theoretical  considera- 
tions, however,  have  not  been  neglected  in  cases  where  they 
have  been  found  necessary  to  fully  explain  a  practical  process, 
and  this  book  is,  therefore,  a  treatise  on  both  the  theory  and 
practice  of  spiral  and  worm  gearing  along  such  lines  as  will 
make  it  especially  useful  to  practical  men. 

Readers  of  mechanical  literature  are  familiar  with  MA- 
CHINERY'S 25-cent  Reference  Books  which  include  the  best  of 
the  material  that  has  appeared  in  MACHINERY  during  the  past 
years,  adequately  revised,  amplified  and  brought  up  to  date. 
Of  these  books,  one  hundred  and  thirty-five  different  titles  have 
been  published  in  the  past  seven  years.  Many  subjects,  how- 

V 

416063 


vi  PREFACE 

ever,  cannot  be  adequately  covered  in  all  their  phases  in  books 
of  this  size,  and  in  answer  to  a  demand  for  more  comprehensive 
and  detailed  treatments  of  the  more  important  mechanical 
subjects,  it  has  been  deemed  advisable  to  bring  out  a  number 
of  larger  volumes,  each  covering  one  subject  completely.  This 
book  is  one  of  these  volumes. 

The'information  contained  in  this  book  is  mainly  compiled  from 
articles  published  in  MACHINERY,  and  the  best  on  the  subject 
that  has  appeared  in  the  Reference  Books  is  also  included,  with 
necessary  modifications  and  additions.  For  the  material  con- 
tained, MACHINERY  is  indebted  to  a  large  number  of  men  who 
have  furnished  practical  information  to  its  columns.  It  has  not 
been  possible  to  give  credit  to  each  individual  contributor  in 
all  instances,  but  it  should  be  mentioned  that  the  framework 
upon  which  the  whole  book  has  been  built  up  consists  of  the 
Reference  Books  and  articles  which  Mr.  Ralph  E.  Flanders,  the 
well-known  gear  expert  and  former  associate  editor  of  MA- 
CHINERY, has  written  and  compiled;  the  chapter  giving  specific 
solutions  for  all  the  different  cases  of  spiral  gear  problems  has 
been  contributed  by  Mr.  J.  H.  Carver;  to  all  other  writers 
whose  material  has  appeared  in  MACHINERY,  and  which  is  now 
used  in  this  book,  the  publishers  hereby  express  their  apprecia- 
tion. 

MACHINERY. 

NEW  YORK,  September,  1914. 


CONTENTS 


CHAPTER  I 

PRINCIPAL  RULES  AND   FORMULAS  FOR  DESIGNING 
SPIRAL  GEARS 

PAGES 

Dimensions  and  Definitions  —  Rules  for  Calculating  Spiral 
Gear  Dimensions  —  Examples  of  Calculations  —  Preliminary 
Graphical  Solution  —  Final  Solution  by  Calculation  —  Vari- 
ations in  the  Methods  Used  —  Basic  Rules  and  Formulas  for 
Spiral  Gears  —  Examples  of  Spiral  Gear  Problems  —  Demon- 
stration of  Grant's  Formula 1-28 

CHAPTER  II 

FORMULAS  FOR  SPECIAL  CASES  OF  SPIRAL  GEAR 

DESIGN 

Procedure  in  Calculating  Spiral  Gears  —  Sixteen  Principal 
Cases  of  Spiral  Gear  Design  with  Formulas  and  Examples  for 
Each  —  Special  Case  of  Spiral  Gear  Design 29-68 

CHAPTER  III 
HERRINGBONE   GEARS 

Definitions  and  Types  of  Herringbone  Gears  —  Cost  of 
Herringbone  Gears  —  Requirements  of  Power  Transmitting 
Mediums  —  Action  of  Spur  Gearing  —  Action  of  Herring- 
bone Gears  —  Advantages  of  Herringbone  Gears  —  Production 
of  Herringbone  Gears  —  One-  and  two-piece  Types  —  Milling 
by  Disk  Cutters  —  Milling  by  End  Mills  —  Planing  Gear 
Teeth  —  The  Hobbing  Process  —  Wuest  Herringbone  Gears 
—  Interchangeable  Herringbone  Gears  —  Width  of  Face  and 
Spiral  Angle  —  Pressure  Angle  —  Tooth  Proportions  —  Diam- 
etral Pitch  —  Pitch  Diameters  and  Center  Distances  —  Gen- 
eral Dimensions  —  Strength  —  Horsepower  Transmitted  — 

vii 


Vlll  CONTENTS 

PAGES 
General  Points  in  Design  —  Summary   of   Salient    Features 

—  Application  to  Steam  Turbines  —  Application  to  Machine 
Tools  —  Application  to  Geared  Hydraulic  Turbines  —  Applica- 
tion to  Rolling  Mills  and  Rod  Mills 69-97 

CHAPTER  IV 

METHODS  FOR  FORMING  THE  TEETH  OF  SPIRAL 
AND  HERRINGBONE  GEARS  AND  WORMS 

Principal  Methods  Used  —  Machines  Using  Formed  Tools 
in  a  Shaping  or  Planing  Operation  —  Machines  Using  Formed 
Milling  Cutters  —  Points  Relating  to  the  Milling  of  Spiral 
Gears  —  Diagram  for  Finding  Cutter  for  Milling  Spiral  Gears 

—  Table  for  Selecting  Cutter  for  Milling  Spiral  Gears  —  An- 
gular Position  of  Table  when  Milling  Spiral  Gears  —  Milling 
the  Spiral  Teeth  —  Specialized  Forms  of  Milling  Machines  for 
Cutting  Spirals  by  the  Formed  Cutter  Method  —  Specialized 
Form  of  Milling  Machines  for  Herringbone  Gears  —  Automatic 
Machines  for  Milling  with  Formed  Cutters  —  The  Molding- 
generating    Principle  —  The    Hobbing    Modification    of    the 
Molding-generating  Principle  —  Field  of  the  Hobbing  Proc- 
ess   for  Helical    Gears  —  Calculating    Gears  for   Generating 
Spirals    on    Hobbing    Machines  —  Universal    Formula    for 
Change  Gears  —  Examples  of  Gear  Calculations  —  Influence  of 
Small  Changes  in  the  Gear  Ratio  on  the  Lead  —  Advantage 
of  the  Differential  Mechanism  on  Gear  Hobbing  Machines  in 
Calculating  Change-gears 98-136 

CHAPTER  V 
HOBS  FOR  SPUR  AND   SPIRAL  GEARS 

Hobbing  vs.  Milling  —  Feed  Marks  Produced  by  Rotating 
Milling  Cutters  —  Comparison  Between  Surfaces  Produced  by 
Milling  and  Hobbing  —  Comparison  of  Output  —  The  Tooth 
Outline  — The  Width  of  Flat  Produced  —  Summary  of  the 
Preceding  Comparative  Study  —  Hobs  for  Spur  and  Spiral 
Gears  —  Causes  of  Defects  in  Hobbed  Gears  —  Grinding  to 
Correct  Hob  Defects  —  Shape  of  Hob  Teeth  —  Diameters  of 
Hobs -J 


CONTENTS  k 

CHAPTER  VI 
CALCULATING   THE   DIMENSIONS   OF  WORM   GEARING 

PAGES 

Definitions  and  Rules  for  Dimensions  of  the  Worm  —  Rules 
for  Dimensioning  the  Worm-wheel  —  Departures  from  the 
Foregoing  Rules  —  Two  Applications  of  Worm  Gearing  — 
Examples  of  Worm  Gearing  Figured  from  the  Rules  —  Di- 
mensions of  Worm-thread  Parts  —  Rules  and  Formulas  for 
the  Design  of  Worm  Gearing  —  Worms  with  Large  Helix 
Angle  —  Table  for  Calculating  the  Outside  Diameter  of 
Worm-wheels  —  Model  Worm-gear  Drawing 156-169 


CHAPTER  VII 

ALLOWABLE  LOAD  AND   EFFICIENCY  OF  WORM 
GEARING 

Relation  of  Load  to  Effort  —  Efficiency  —  Allowable  Load 
—  Relation  Between  Velocity  at  Pitch  Line,  Angle  of  Thread 
and  Efficiency  —  German  Experiments  to  Determine  Speed 
Factor  —  Safe  Load  on  Worm-gear  Teeth  —  Practical  Points 
in  the  Design  of  Worm  and  Gear  —  Self-locking  Worm  Gear- 
ing —  An  Example  from  Practice  —  Theoretical  Efficiency  of 
Worm  Gearing  —  Worm  and  Helical  Gears  as  Applied  to 
Automobile  Rear  Axle  Drives  —  Worm  Gearing  Employed 
for  Freight  Elevators  —  Frequently  Employed  Objectionable 
Designs  —  Lubricants  for  Worm-gears  —  Horsepower  of  Worm 
Gearing 170-190 


CHAPTER  VIII 

THE  DESIGN  OF  SELF-LOCKING  WORM-GEARS 

Conditions  under  which  Worm  Gearing  Becomes  Self-lock- 
ing —  Bearing  Friction  —  Table  giving  Moment  of  Friction 
with  Various  Types  of  Bearings  —  General  Method  of  Pro- 
cedure of  Calculation 191-201 


X  CONTENTS 

CHAPTER  IX 

HINDLEY  WORM  AND   GEAR  PAGES 

Comparison  of  Ordinary  and  Hindley  Worm  Gearing  — 
Nature  of  Contact  of  Hindley  Worm  Gearing  —  Considera- 
tions of  the  Ideal  Case  —  Objections  to  the  Hindley  Gear  — 
Modifications  of  Hindley  Worm-gear  Practice  —  Conclusions 
regarding  the  Hindley  Worm  and  Gear 202-211 

CHAPTER  X 

METHODS   FOR  FORMING  THE  TEETH  OF  WORM- 
,     WHEELS 

Gashing  Worm-wheels  by  the  Formed  Cutter  Process  — 
The  Molding-generating  Principle  —  Hobbing  Worm-wheels 
in  the  Milling  Machine  —  Gearing  for  Worm-wheel  Hobbing 
Machines  —  The  Fly-tool  Method  of  Cutting  Worm-wheels  — 
Taper  Hob  for  Cutting  Worm-wheels  —  Ordinary  and  Taper 
Hob  Method  of  Hobbing  Worm-wheels  Compared  —  Effi- 
ciency of  Taper-hobbed  Worm  Gearing  —  The  Various  Meth- 
ods Compared  —  Points  relating  to  the  Worm  —  Manufac- 
ture of  Hindley  Worm  Gearing 212-230 

CHAPTER  XI 
GASHING  AND  HOBBING  A  WORM-WHEEL 

The  Gashing  and  Hobbing  Process  —  Setting  the  Worm  in 
the  Machine  —  Setting  the  Table  of  the  Machine  —  The 
Gashing  Operation  —  The  Hobbing  Operation  —  Reducing 
Flats  on  Hobbed  Worm-wheel  Teeth  —  Suggested  Refinement 
in  the  Hobbing  of  Worm-wheels 231-243 

CHAPTER  XII 
HOBS  FOR  WORM-GEARS 

Dimensions  of  Hobs  —  Thread  Tool  for  the  Hob  —  Char- 
acter of  Flutes  —  Spiral  Fluted  Hob  Angles  —  Graphical 
Method  for  Determining  Angle  of  Flute  —  Lengths  of  Worms 
and  Hobs  —  Number  of  Flutes  in  Hobs  — Imperfect  Generat- 
ing Action  of  the  Hob  —  Diagrams  for  Finding  the  Number 
of  Cuts  per  Linear  Inch  —  The  Effect  of  the  Number  of  Teeth 
in  the  Wheel  —  General  Formula  for  Determining  the  Num- 
ber of  Flutes  —  Hobbing  Methods  which  give  a  Complete 
Generating  Action 244-265 


SPIRAL    AND    WORM    GEARING 


CHAPTER  I 

PRINCIPAL  RULES  AND   FORMULAS  FOR  DESIGNING 
SPIRAL  GEARS 

THE  subject  of  spiral  or  helical  gearing  is  one  which,  from  its 
very  nature,  can  be  approached  by  any  one  of  a  number  of  differ- 
ent ways,  and  it  has  been  approached  by  so  many  of  these  possible 
different  ways  that  the  subject  has,  perhaps,  become  quite  con- 
fused in  the  minds  of  many  readers  of  technical  literature. 

The  terms  " spiral  gear"  and  " helical  gear"  are,  in  usage, 
synonymous,  but  the  former  of  these  terms  is,  theoretically,  in- 
correct. Inasmuch,  however,  as  the  word  " spiral"  is  in  such 
common  use  among  mechanics  in  this  connection,  it  has  been 
used  freely  throughout  this  treatise. 

Dimensions  and  Definitions.  —  Some  of  the  terms  used  will 
require  explanation.  The  center  angle  of  a  pair  of  helical  or 
spiral  gears  is  the  angle  made  by  the  two  center  lines  or  axes  of 
the  gears,  as  taken  in  a  view  perpendicular  to  both  axes.  In 
Fig.  i  are  shown  views  of  three  sets  of  spiral  gears  taken  in  the 
plane  which  shows  the  center  angle.  At  the  left  is  the  ordinary 
case  in  which  the  shafts  are  at  right  angles  with  each  other,  so 
that  the  center  angle  (7)  is  90  degrees.  In  the  second  case  7 
is  less  than  90  degrees,  and  in  the  example  shown  at  the  right  it 
is  more.  It  should  be  noted  in  the  last  two  cases  that  the  posi- 
tion of  the  shaft  axes  is  identical,  but  that  the  two  center  angles 
are  located  on  opposite  sides  of  axis  A.  In  order  to  know  on 
which  side  of  the  center  line  to  take  the  center  angle  in  cases  like 
those  shown,  we  have  to  reckon  with  the  position  of  the  teeth 
of  the  gears  in  contact.  The  center  angle  is  taken  at  the  side 
which  includes  the  line  x-x,  passing  lengthwise  of  the  teeth  of 


SPIRAL  GEARING 


the  gears  at  the  point  of  contact  with  each  other.  Since  the 
teeth  are  laid  out  differently  in  the  two  cases,  the  angles  are 
different.  The  case  shown  in  the  center  is  the  more  usual  of 
the  two,  the  other  being  very  rare. 

In  Fig.  2  is  given  a  diagram  showing  what  is  meant  by  the 
" tooth  angle"  of  a  helical  gear.  In  using  the  expression  " tooth 
angle,"  the  angle  made  by  the  tooth  with  the  axis  of  the  gear 
is  meant,  not  the  angle  of  the  tooth  with  the  face  of  the  gear. 
Fig.  2  shows  aa  as  the  tooth  angle  of  gear  a,  and  ab  as  the  tooth 
angle  of  gear  b,  used  in  the  sense  in  which  we  will  use  them. 

The  number  of  teeth  and  the  pitch  diameter  are  terms  which 
are  identically  the  same  as  those  used  for  spur  gearing  and, 


Machinery 


Fig.  i.     Spiral  Gears  with  Different  Center  Angles 

therefore,  require  no  explanation  in  this  connection,  it  being  a 
necessary  assumption  that  anyone  attempting  to  design  a  pair 
of  spiral  gears  is  well  familiar  with  the  design  of  spur  gears. 
Practically  all  spiral  gears  are  of  small  size,  and  hence  are  reck- 
oned on  the  diametral  pitch  rather  than  the  circular  pitch  sys- 
tem. All  the  rules  and  formulas  given  will,  therefore,  make 
use  of  the  diametral  pitch  only.  This  may  easily  be  found  from 
the  circular  pitch  by  dividing  3.1416  by  the  circular  pitch.  The 
center  distance  is,  of  course,  the  shortest  distance  between  the 
axes,  and  so  is  measured  along  the  perpendicular  common  to 
both  of  them. 

The  regular  diametral  pitch  of  a  spiral  gear  will  be  found,  the 
same  as  for  a  spur  gear,  by  dividing  the  number  of  teeth  by  the 
pitch  diameter  in  inches.  We  are  not  interested  in  knowing 


RULES  AND   FORMULAS 


what  this  is,  however,  since  it  does  not  enter  into  the  calcula- 
tions, and  since  the  cutter  used  has  to  be  for  a  somewhat  finer 
diametral  pitch.  This  is  shown  more  clearly  in  Fig.  3.  The 
normal  diametral  pitch,  or  diametral  pitch  of  the  cutter  used,  is 
reckoned  from  measurements  taken  along  the  pitch  cylinder  at 
right  angles  to  the  length  of  the  tooth.  Pf  represents  the  regu- 
lar circular  pitch,  while  Pnr  represents  the  normal  circular  pitch. 
The  diametral  pitch  may  be  found  from  this  by  dividing  3.1416 
by  Pn'.  This  is  the  pitch  of  the  cutter  to  be  used.  The  cutter, 
as  will  be  explained  in  the  following,  cannot  be  selected  for  the 


Machinery 


Fig.  2.     Diagram  showing  Notation  used  for  Tooth  Angles 

actual  number  of  teeth  in  the  gear,  but  must  take  into  account 
the  helix  angle  of  the  teeth  as  well,  since  the  curvature  as  meas- 
ured on  a  line  at  right  angles  to  the  helix  is  at  a  greater  radius 
than  when  measured  on  the  circle. 

The  length  of  the  helix,  or  the  lead,  as  shown  in  Fig.  3,  is  the 
length  of  pitch  cylinder  required  to  permit  one  complete  revolu- 
tion of  the  tooth  if  the  latter  were  carried  around  for  the  full 
length  of  this  cylinder.  In  Fig.  4  the  relation  of  lead,  circum- 
ference and  tooth  angle  is  plainly  shown,  the  helix  AB  here 
being  developed  on  a  plane.  The  addendum  5,  and  whole  depth 
W  of  the  tooth  for  helical  gears,  is  the  same  as  for  plain  spur 


SPIRAL  GEARING 


gears.  The  normal  thickness  of  tooth  at  the  pitch  line  Tn,  as 
shown  in  Fig.  3,  is  measured  in  a  direction  perpendicular  to  the 
face  of  the  tooth.  The  regular  tooth  thickness  is  shown  at  T\ 
this,  however,  does  not  enter  into  the  calculations.  The  out- 
side diameter,  as  for  spur  gears,  is  found  by  adding  twice  the 
addendum  to  the  pitch  diameter. 


Machinery 


Fig.  3. 


Diagram  of  Spiral  Gear  Illustrating  Terms  used  in  the 
Calculations 


Rules  for  Calculating  Spiral  Gear  Dimensions.  —  The  follow- 
ing rules  are  used  for  calculating  the  dimensions  of  spiral  or 
helical  gears: 

Rule  i .  The  sum  of  the  tooth  angles  of  a  pair  of  mating  heli- 
cal gears  is  equal  to  the  shaft  angle;  that  is  to  say,  in  Figs,  i 
and  2  angle  aa  added  to  ab  equals  7,  as  is  self-evident  from  the 
engravings. 

Rule  2.  To  find  the  pitch  diameter  of  a  helical  gear,  divide  the 
number  of  teeth  by  the  product  of  the  normal  pitch  and  the 
cosine  of  the  tooth  angle. 

Rule  3.  To  find  the  center  distance,  add  together  the  pitch 
diameters  of  the  two  gears  and  divide  by  2.  This  rule  is  evi- 
dently the  same  as  for  spur  gears. 

Rule  4.  To  prove  the  calculations  for  pitch  diameters  and 
center  distance,  multiply  the  number  of  teeth  in  the  first  gear 
by  the  tangent  of  the  tooth  angle  of  that  gear,  and  add  the  num- 


RULES  AND   FORMULAS 


her  of  teeth  in  the  second  gear  to  the  product;  the  sum  should 
equal  twice  the  product  of  the  center  distance  multiplied  by  the 
normal  diametral  pitch,  multiplied  by  the  sine  of  the  tooth 
angle  of  the  first  gear. 

Rule  5.  To  find  the  number  of  teeth  for  which  to  select  the 
cutter,  divide  the  number  of  teeth  in  the  gear  by  the  cube  of 
the  cosine  of  the  tooth  angle. 


Machinery 


Fig.  4. 


Diagram  showing  Relation  between  Pitch  Diameter,  Lead 
and  Helix  Angle 


Rule  6.  To  find  the  lead  of  the  tooth  helix,  multiply  the  pitch 
diameter  by  3.1416  times  the  cotangent  of  the  tooth  angle. 

The  rules  relating  to  the  addendum  and  the  whole  depth  of 
tooth  are  the  same  as  for  spur  gears.  They  are: 

Rule  7.  To  find  the  addendum,  divide  i  by  the  normal  diam- 
etral pitch. 

Rule  8.  To  find  the  whole  depth  of  tooth  space,  divide  2.157 
by  the  normal  diametral  pitch. 


6  SPIRAL  GEARING 

Rule  9.  To  find  the  normal  tooth  thickness  at  the  pitch  line, 
divide  1.571  by  the  normal  diametral  pitch. 

Rule  10.  To  find  the  outside  diameter,  add  twice  the  adden- 
dum to  the  pitch  diameter. 

The  problem  of  designing  a  pair  of  spiral  gears  presents  itself 
in  general  in  two  different  forms  or  classes,  which  may  be  stated 
as  follows: 

Class  i.  The  diametral  pitch  and  the  numbers  of  teeth  in 
the  two  gears  are  given. 

Class  2.  A  fixed  center  distance  is  given  together  with  the 
velocity  ratio  or  the  numbers  of  teeth,  witfh  the  requirement 
that  standard  cutters  of  even  diametral  pitch  be  used. 

Examples  of  Calculations  Under  Class  i.  —  Let  it  be  required 
to  make  the  necessary  calculations  for  a  pair  of  spiral  gears  in 
which  the  shafts  are  at  right  angles.  Normal  diametral  pitch 
equals  3;  number  of  teeth  in  gear  equals  45;  number  of  teeth 
in  pinion  equals  18. 

There  being  no  restriction  in  this  particular  case  as  to  center 
distance,  we  have  to  settle  first  on  the  tooth  angles  for  the  two 
gears.  To  obtain  the  highest  efficiency,  some  authorities  advise 
that  the  smallest  tooth  angle  be  given  to  the  gear  having  the 
smallest  number  of  teeth;  and  this  angle  should  not,  in  general, 
run  below  20  degrees.  Keeping  it  nearly  30  or  even  up  to  45 
would  be  better.  On  the  basis  aa  =  30  and  ab  =  60  degrees, 
we  have  the  following  calculations: 

To  find  the  pitch  diameters,  use  Rule  2 : 

Pitch  diameter  of  gear      = }       0  =  30  inches.     . 

Pitch  diameter  of  pinion  =  -  — z  =  6.028  inches. 

3  X  cos  30° 

To  find  the  center  distance,  use  Rule  3: 
3°  +  6'928  =  I8.464  inches. 

2 

To  prove  that  the  previous  calculations  are  correct,  use  Rule  4 : 

45  X  tan  60°  +  18  =  95.940. 
2  X  18.464  X  3  X  sin  60°  =  95.939. 


RULES  AND  FORMULAS  7 

These  two  results  are  so  nearly  alike  that  the  previous  cal- 
culations may  be  considered  fully  correct. 

To  find  the  number  of  teeth  for  which  to  select  the  cutter,  use 
Rule  5: 

Forgear> 


For  pinion,  r—   —  ^  =  28,  approximately. 

To  find  the  lead  of  the  tooth  helix,  use  Rule  6: 

Lead  for  gear     =  3.1416  X  30       X  cot  60°  =  54.38  inches. 

Lead  for  pinion  =  3.1416  X  6.928  X  cot  30°  =  37.70  inches. 

To  find  the  addendum,  use  Rule  7  : 

Addendum  =  ^  =  0.333  inch- 

To  find  the  whole  depth  of  tooth  space,  use  Rule  8: 

Whole  depth  =  2>I57  =  0.719  inch. 

o 

To  find  the  normal  tooth  thickness  at  the  pitch  line,  use 
Rule  9: 


Tooth  thickness  =    ''    =0.523  inch. 
o 

To  find  the  outside  diameter,  use  Rule  10: 

For  gear,     30       +  0.666  =  30.666  inches. 

For  pinion,  6.928  +  0.666  =  7.594  inches. 

This  concludes  the  calculations  for  this  example.  If  it  is  re- 
quired that  the  pitch  diameters  of  both  gears  be  more  nearly 
alike,  the  tooth  angle  of  the  gear  must  be  decreased,  and  that  of 
the  pinion  increased. 

Suppose  we  have  a  case  in  which  the  requirements  are  the  same 
as  in  Example  i,  but  it  is  required  that  both  gears  shall  have 
the  same  tooth  angle  of  45  degrees.  Under  these  conditions  the 
addendum,  whole  depth  of  tooth  and  normal  thickness  at  the 
pitch  line  would  be  the  same,  but  the  other  dimensions  would 
be  altered  as  below: 

Pitch  diameter  of  gear  =  -  **  -  3  =  21.216  inches. 

3  X  cos  45 

Pitch  diameter  of  pinion  =  -  5  =  8.487  inches. 

3  X  cos  45° 


8  SPIRAL  GEARING 

o  ,.  ,  21.216+8.487  0       .     , 

Center  distance  =  —  — L  =  14.851  inches. 

Number  of  teeth  for  which  to  select  cutter: 

For  gear,  - —  ^  0    •=  127,  approximately, 
(cos  45  ; 

For  pinion.  -, -^-z  =51,  approximately. 

(cos  45  )3 

Lead  of  helix  for  gear  =  3.1416  X  21.216  X  cot  45° 

=  66.65  inches. 

Lead  of  helix  for  pinion  =  3.1416  X  8.487  X  cot  45° 

=  26.66  inches. 

Outside  diameter  of  gear      =  21.216  +  0.666  =  21.882  inches. 

Outside  diameter  of  pinion  =    8.487  +  0.666  =    9.153  inches. 

Examples  of  Calculations  Under  Class  2.  —  In  Class  2  use  is 
made  of  the  term  "equivalent  diameter."  The  quotient  ob- 
tained by  dividing  the  number  of  teeth  in  a  helical  gear  by  the 
diametral  pitch  of  the  cutter  used  gives  us  a  very  useful  factor 
for  figuring  the  dimensions  of  helical  gears,  and  this  has  been 
given  the  name  "equivalent  diameter,"  an  abbreviation  of  the 
words  "diameter  of  equivalent  spur  gear,"  which  more  accu- 
rately describe  it.  This  quantity  cannot  be  measured  on  the 
finished  gear  with  a  rule,  being  only  an  imaginary  unit  of  meas- 
urement. 

Rule  ii.  To  find  the  equivalent  diameter  of  a  helical  gear, 
divide  the  number  of  teeth  of  the  gear  by  the  diametral  pitch  of 
the  cutter  by  which  it  is  cut. 

Preliminary  Graphical  Solution.  —  The  process  of  locating  a 
railway  line  over  a  mountain  range  is  divided  into  two  parts: 
the  preliminary  survey  or  period  of  exploration,  and  the  final 
determination  of  the  grade  line.  The  problem  of  designing  a 
pair  of  helical  gears  resembles  this  engineering  problem  in  having 
many  possible  solutions,  from  which  it  is  the  business  of  the 
designer  to  select  the  most  feasible.  For  the  exploration  or  pre- 
liminary survey  the  diagram  shown  in  Fig.  5  will  be  found  a 
great  convenience.  The  materials  required  are  a  ruler  with  a 
good  straight  edge,  and  a  piece  of  accurately  ruled,  or,  prefer- 


RULES  AND   FORMULAS 


ably,  engraved,  cross-section  paper.  If  a  point  O  be  so  located 
on  the  paper  that  BO,  the  distance  to  one  margin  line,  be  equal 
to  the  equivalent  diameter  of  gear  a,  while  B'O,  the  distance  to 
the  other  margin  line,  be  equal  to  the  equivalent  diameter  of 
gear  6,  then  (when  the  rule  is  laid  diagonally  across  the  paper  in 
any  position  that  cuts  the  margin  lines  and  passes  through  point 
0)  DO  will  be  the  pitch  diameter  of  gear  a,  D'O  the  pitch  diam- 
eter of  gear  b,  angle  BOD  the  tooth  angle  of-  gear  a  and  angle 
B'ODf  the  tooth  angle  of  gear  b.  This  simple  diagram  presents 


Machinery 


Fig.  5.     Preliminary  Solution  with  a  Rule  and  Cross-section  Paper 

instantly  to  the  eye  all  possible  combinations  for  any  given 
problem.  It  is,  of  course,  understood  that  in  the  shape  shown 
it  can  only  be  used  for  shafts  making  an  angle  of  90  degrees 
with  each  other. 

The  diagram  as  illustrated  shows  that  a  pair  of  helical  gears 
having  12  and  21  teeth  each,  cut  with  a  5-pitch  cutter,  and  hav- 
ing shafts  at  90  degrees  with  each  other  and  5  inches  apart, 
may  have  tooth  angles  of  36°  52'  and  53°  8',  and  pitch  diameters 
of  3  inches  and  7  inches,  respectively. 


10  SPIRAL  GEARING 

Suppose  it  were  required  to  figure  out  the  essential  data  for 
three  sets  of  helical  gears  with  shafts  at  right  angles,  as  follows: 

i  st.  Velocity  ratio  2  to  i,  center  distance  between  shafts 
2\  inches. 

2d.  Velocity  ratio  2  to  i,  center  distance  between  shafts 
3!  inches. 

3d.  Velocity  ratio  2  to  i,  center  distance  between  shafts 
4  inches. 

We  will  take  the  first  of  these  to  illustrate  the  method  of  pro- 
cedure about  to  be  described. 

We  have  a  center  distance  of  2\  inches  and  a  speed  ratio  be- 
tween driver  and  driven  shafts  of  2  to  i.  The  first  thing  to 
determine  is  the  pitch  of  the  cutter  to  use.  The  designer  selects 
this  according  to  his  best  judgment,  taking  into  consideration 
the  cutters  on  hand  and  the  work  the  gearing  will  have  to  do. 
Suppose  he  decides  that  i2-pitch  will  be  about  right.  In  Fig.  5 
it  will  be  remembered  that  DO  was  the  pitch  diameter  of  gear  a, 
while  D'O  was  the  pitch  diameter  of  gear  b.  That  being  the 
case  DOD'  is  equal  to  twice  the  distance  between  the  shafts. 
In  the  problem  under  consideration  this  will  be  equal  to  2  X  2}, 
or  4!  inches.  Fig.  6  is  a  skeleton  outline  showing  the  operation 
of  making  the  preliminary  survey  with  rule  and  cross-section 
paper.  AG  and  AGr  represent  the  margin  lines  of  the  sheet, 
while  DD'  represents  the  graduated  straight-edge.  By  the  con- 
ditions of  the  problem,  the  distance  between  points  D  and  D', 
where  the  ruler  crosses  the  margin  lines,  must  be  equal  to  4^ 
inches.  There  has  next  to  be  determined  at  what  angle  of  in- 
clination the  ruler  shall  be  placed  in  locating  this  line.  To  do 
this  we  will  first  find  our  "ratio  line."  Select  any  point  C 
such  that  CF'  is  to  CF  as  2  is  to  i,  which  is  the  required  ratio  of 
the  gears.  Draw  through  point  C,  so  located,  the  line  AE. 
Line  AE  is  then  the  ratio  line,  that  is,  a  line  so  drawn  that  the 
measurements  taken  from  any  point  on  it  to  the  margin  lines 
will  be  to  each  other  in  the  same  ratio  as  the  required  ratio  be- 
tween the  driving  and  driven  gear.  Now,  by  shifting  the  ruler 
on  the  margin  lines,  always  being  careful  that  they  cut  off  the 
required  distance  of  4^  inches  on  the  graduations,  it  is  found 


RULES  AND   FORMULAS 


II 


that  when  the  rule  is  laid  as  shown  in  position  No.  i,  cutting  the 
ratio  line  at  0',  the  distance  from  the  point  of  intersection  to 
corner  A  is  at  its  maximum.  For  the  minimum  value  the  tooth 
angle  is  the  limiting  feature.  For  a  gear  of  this  kind  30  degrees 
is,  perhaps,  about  as  small  as  would  be  advisable,  so  when  the 
ruler  is  inclined  at  an  angle  of  about  30  degrees  with  margin 
line  AG',  and  occupies  position  No.  2  as  shown,  it  will  cut  line 
AE  at  0" ',  and  the  distance  cut  off  from  the  point  of  intersection 


Machinery 


Fig.  6.     Preliminary  Graphical  Solution  for  Problem  No.  i 

to  corner  A  will  be  at  its  minimum  value.  The  ruler  must  then 
be  located  at  some  intermediate  position  between  No.  i  and 
No.  2. 

Supposing,  for  example,  14  teeth  in  gear  a  and  28  teeth  in  gear 
b  be  tried.  According  to  Rule  n  the  equivalent  diameter  of 
gear  a  will  then  be  14-^-12,  or  1.1666  inch;  the  equivalent 
diameter  of  b  will  be  28  -f-  12,  or  2.3333  inches.  Returning  to 
the  diagram  to  locate  the  point  of  intersection,  it  will  be  found 
that  point  0"f  is  so  located  that  lines  drawn  from  it  to  AG  and 


12  SPIRAL  GEARING 

AG'  will  be  equal  to  1.1666  inch  and  2.3333  inches,  respectively, 
but  this  is  beyond  point  Of,  which  was  found  to  be  the  outermost 
point  possible  to  intersect  with  a  4^-inch  line  DD' '.  This  shows 
that  the  conditions  are  impossible  of  fulfillment. 

Trying  next  12  teeth  and  24  teeth,  respectively,  for  the  two 
gears,  the  equivalent  diameters  by  Rule  n  will  be  i  inch  and 
2  inches.  Point  0  is  now  so  located  that  OB  equals  i  inch  and 
OB'  equals  2  inches.  Seeing  that  this  falls  as  required  between 
0'  and  0" ',  stick  a  pin  in  at  this  point  to  rest  -the  straight-edge 
against,  and  shift  the  straight-edge  about  until  it  is  located  in 
such  an  angular  position  that  the  margin  lines  AG  and  AG' 
cut  off  4^  inches,  or  twice  the  required  distance  between  the 
shafts,  on  the  graduations.  This  gives  the  preliminary  solution 
to  the  problem.  Measuring  as  carefully  as  possible,  DO,  the  pitch 
diameter  of  gear  a,  is  found  to  be  about  1.265  mcn  diameter, 
and  D'O,  the  pitch  diameter  of  gear  b,  about  3.235  inches.  Angle 
BOD,  the  tooth  angle  of  gear  a,  measures  about  37°  50'.  Angle 
B'OD',  the  tooth  angle  of  gear  b,  would  then  be  52°  10'  accord- 
ing to  Rule  i. 

Final  Solution  by  Calculations.  —  To  determine  angle  BOD 
more  accurately  than  is  feasible  by  a  graphical  process,  use  the 
following  rule: 

Rule  12.  The  tooth  angle  of  gear  a  in  a  pair  of  mating  helical 
gears  a  and  b,  whose  axes  are  90°  apart,  must  be  so  selected 
that  the  equivalent  diameter  of  gear  b  plus  the  product  of  the 
tangent  of  the  tooth  angle  of  gear  a  by  the  equivalent  diameter 
of  gear  a  will  be  equal  to  the  product  of  twice  the  center  dis- 
tance by  the  sine  of  the  tooth  angle  of  gear  a.  (This  rule,  it  will 
be  seen,  is  simply  a  modification  of  Rule  4.) 

That  is  to  say  in  this  case,  OB'  +  (OB  X  the  tangent  of  angle 
BOD)  =  DD'  X  the  sine  of  angle  BOD.  Perform  the  opera- 
tions indicated,  using  the  dimensions  which  were  derived  from 
the  diagram,  to  see  whether  the  equality  expressed  in  this  equa- 
tion holds  true.  Substituting  the  numerical  values: 
2  +  (i  X  0.77661)  =  4.5  X  0.61337, 

2.77661  =  2.76016, 
a  result  which  is  evidently  inaccurate. 


RULES  AND  FORMULAS  13 

The  solution  of  the  problem  now  requires  that  other  values 
for  angle  BOD,  slightly  greater  or  less  than  37°  50',  be  tried 
until  one  is  found  that  will  bring  the  desired  equality.  It  will  be 
found  finally  that  if  the  value  of  38°  20'  be  used  as  the  tooth 
angle  of  gear  a,  the  angle  is  as  nearly  right  as  one  could  wish. 
Working  out  Rule  12  for  this  value: 

2  +  (i  X  0.79070)  =  4.5  X  0.62024 
2.79070     =  2.79108. 

This  gives  a  difference  of  only  0.00038  between  the  two  sides  of 
the  equation.  The  final  value  of  the  tooth  angle  of  gear  a  is 
thus  settled  as  being  equal  to  38°  20'.  Applying  Rule  i  to  find 
the  tooth  angle  of  gear  b  we  have:  90°  —  38°  20'  =  51°  40'. 
The  next  rule  relates  to  finding  the  pitch  diameter  of  the  gears. 

Rule  13.  The  pitch  diameter  of  a  helical  gear  equals  the 
equivalent  diameter  divided  by  the  cosine  of  the  tooth  angle 
(or  the  equivalent  diameter  multiplied  by  the  secant  of  the  tooth 
angle).  This  rule  is  a  modification  of  Rule  2. 

If  a  table  of  secants  is  at  hand,  it  will  be  somewhat  easier  to 
use  the  second  method  suggested  by  the  rule,  since  multiplying 
is  usually  easier  than  dividing.  Using  in  this  case,  however, 
the  table  of  cosines,  and  performing  the  operation  indicated  by 
Rule  13,  we  have  for  the  pitch  diameter  of  gear  a: 

1  -f-  0.78442  =  1.2748,  or  1.275  incn>  nearly; 
and  for  the  pitch  diameter  of  gear  b : 

2  -T-  0.62024  =  3.2245,  or  3.225  inches,  nearly. 

To  check  up  the  calculations  thus  far,  the  pitch  diameter  of  the 
two  gears  thus  found  may  be  added  together.  The  sum  should 
equal  twice  the  center  distance,  thus: 

1.275  +3.225  =  4.500 
which  proves  the  calculations  for  the  angle. 

Applying  Rule  10  to  gear  a: 

1.2748  +  (2  -i-  12)  =  1.2748  +  0.1666  =  1.4414  =  1.441  inch, 
nearly. 

For  gear  b: 

3.2245  +  (2  -5-  12)  =  3.2245  +  0.1666  =  3.3911  =  3-391  inches, 
nearly. 


14  SPIRAL  GEARING 

In  cutting  spur  gears  of  any  given  pitch,  different  shapes  of 
cutters  are  used,  depending  upon  the  number  of  teeth  in  the 
gear  to  be  cut.  For  instance,  according  to  the  Brown  &  Sharpe 
system  for  involute  gears,  eight  different  shapes  are  used  for 
cutting  the  teeth  in  all  gears,  from  a  i2-tooth  pinion  to  a  rack. 
The  fact  that  a  certain  cutter  is  suited  for  cutting  a  i2-tooth 
spur  gear  is  no  sign  that  it  is  suitable  for  cutting  a  i2-tooth 
helical  gear,  since  the  fact  that  the  teeth  are  cut  on  an  angle 
alters  their  shape  considerably.  To  find  out  the  number  of 
teeth  for  which  the  cutter  should  be  selected,  use  Rule  5. 

Applying  Rule  5  to  gear  a: 

12  -r-  O.7843  =  12  -f-  0.4818  =  25  — 
and  for  gear  b: 

24  -T-  O.62O3  =  24-7-  0.2383  =  100  + 

giving,  according  to  the  Brown  &  Sharpe  system,  cutter  No.  5 
for  gear  a  and  cutter  No.  2  for  gear  b. 

In  gearing  up  the  head  of  the  milling  machine  to  cut  these 
gears  it  is  necessary  to  know  the  lead  of  the  helix  or  "spiral" 
required  to  give  the  tooth  the  proper  angle.  To  find  this  use 
Rule  6.  In  solving  problems  by  this  rule,  as  for  Rule  5,  it  will 
be  sufficient  to  use  trigonometrical  functions  to  three  significant 
places  only,  this  being  accurate  enough  for  all  practical  purposes. 
Solving  by  Rule  6  to  find  the  lead  for  which  to  set  up  the  gearing 
in  cutting  a: 

1.275  X  1.265  X  3.14  =  5-065,  or  5^  inches,  nearly; 
for  gear  b: 

3.225  X  0.79 1  X  3.14  =  8.010,  or  8^-  inches,  nearly. 

The  lead  of  the  helix  must  be,  in  general,  the  adjustable 
quantity  in  any  spiral  gear  calculation.  If  special  cutters  are 
to  be  made,  the  lead  of  the  helix  may  be  determined  arbitrarily 
from  those  given  in  the  milling  machine  table;  this  will,  however, 
probably  necessitate  a  cutter  of  fractional  pitch.  On  the  other 
hand,  by  using  stock  cutters  and  varying  the  center  distance 
slightly,  we  might  find  a  combination  which  would  give  us  for 
one  gear  a  lead  found  in  the  milling  machine  table,  but  it  would 
only  be  chance  that  would  make  the  lead  for  the  helix  in  the 


RULES  AND  FORMULAS  15 

mating  gear  also  of  standard  length.  It  is  then  generally  better 
to  calculate  the  milling  machine  change  gears  according  to  the 
usual  methods  to  suit  odd  leads,  rather  than  to  adapt  the  other 
conditions  to  suit  an  even  lead.  It  will  be  found  in  practice 
that  the  lead  of  the  helix  may  be  varied  somewhat  from  that 
calculated  without  seriously  affecting  the  efficiency  of  the  gears. 

The  remaining  calculations  relating  to  the  proportions  of  the 
teeth  do  not  vary  from  those  for  spur  gears  and  are  here  set 
down  for  the  sake  of  completeness  only. 

The  addendum  of  a  standard  gear  is  found  by  Rule  7 : 

For  gears  a  and  b  this  will  give: 

i  -T-  12  =  0.0833  mcn- 

The  whole  depth  of  the  tooth  is  found  by  Rule  8: 

This  gives  for  gears  a  and  b: 

2.157  -T-  12  =  0.1797  inch. 

The  thickness  of  the  tooth  is  found  by  Rule  9: 

For  gears  a  and  b  of  the  problem  this  gives: 
1.571  -r-  12  =  0.1309  inch. 

Variations  in  Methods  Used.  —  This  completes  all  the  cal- 
culations required  to  give  the  essential  data  for  making  our  first 
pair  of  helical  gears.  To  illustrate  the  variety  of  conditions  for 
which  these  problems  may  be  solved,  the  other  cases  will  be 
worked  out  somewhat  differently.  In  the  case  just  considered 
no  allowance  was  made  for  possible  conditions  which  might  have 
limited  the  dimensions  of  the  gears,  and  the  problem  was  solved 
for  what  might  be  considered  good  general  practice.  Gear  a, 
however,  might  have  been  too  small  to  put  on  the  shaft  on  which 
it  was  intended  to  go,  while  gear  b  might  have  been  too  large 
to  enter  the  space  available  for  it.  If,  as  we  may  assume,  these 
gears  are  intended  to  drive  the  cam-shaft  of  a  gas  engine,  the 
solution  would  probably  be  unsatisfactory.  Case  No.  2  will 
therefore  be  solved  for  a  center  distance  of  3!  inches  as  required, 
but  the  two  gears  will  be  made  of  about  equal  diameter. 

Fig.  7  shows  the  preliminary  graphical  solution  of  this  problem, 
the  reference  letters  in  all  cases  being  the  same  as  in  Fig.  6. 
With  a  lo-pitch  cutter,  if  this  suited  the  judgment  of  the  designer, 


i6 


SPIRAL  GEARING 


15  teeth  in  gear  a  and  30  teeth  in  gear  b  would  require  that  the 
point  of  intersection  on  the  ratio  line  AE  be  located  at  0  where 
BO  equals  the  equivalent  diameter  of  gear  a,  which  equals  ij 
inch,  while  B'O  equals  the  equivalent  diameter  of  gear  b,  or 


Machinery 


Fig.  7.     Solution  of  Problem  No.  2  for  Equal  Diameters 

3  inches,  both  calculated  in  accordance  with  Rule  n.  The  re- 
quired condition  now  is  that  DO  be  approximated  to  D'0\  that 
is  to  say,  that  the  pitch  diameters  of  the  two  gears  be  about 
equal.  After  continued  trial  it  will  be  found  impossible  to  locate 
0,  using  a  cutter  of  standard  diametral  pitch,  so  that  DO  and 


RULES  AND   FORMULAS  17 

D'O  shall  be  equal,  and  at  the  same  time  have  DD'  equal  to 
twice  the  required  center  distance,  which  is  2  X  3!  inches  or 
6f  inches.  If  this  center  distance  could  be  varied  slightly  with- 
out harm,  BD  could  be  taken  as  equal  to  AB\  then  it  would  be 
found  that  a  line  drawn  from  D  through  O  to  D' ',  though  giving 
a  somewhat  shortened  center  distance,  would  make  two  gears 
of  exactly  the  same  pitch  diameter. 

Drawing  line  DOD',  however,  as  first  described  to  suit  the 
conditions  of  the  problem,  and  measuring  it  for  a  preliminary 
solution  the  following  results  are  obtained:  The  tooth  angle  of 
gear  a  =  angle  BOD  =  63°  45';  and  the  tooth  angle  of  gear  b  = 
angle  B'OD'  =  90°  -  63°  45'  =  26°  15',  according  to  Rule  i. 
Performing  the  operations  indicated  in  Rule  12  to  correct  these 
angles,  it  is  found  that  when  the  tooth  angle  of  gear  a  is  63°  54', 
and  that  for  gear  b  is  26°  6',  the  equation  of  Rule  12  becomes: 

3  +  (15  X  2.04125)  =  6.75  X  0.89803 
6.06187  =  6.06170 

which  is  near  enough  for  all  practical  purposes.  The  other  dimen- 
sions are  easily  obtained  as  before  by  using  the  remaining  rules. 

To  still  further  illustrate  the  flexibility  of  the  helical  gear 
problem,  the  third  case,  for  a  center  distance  of  4  inches,  will  be 
solved  in  a  third  way.  It  is  shown  in  MacCord's  " Kinematics" 
that  to  give  the  least  amount  of  sliding  friction  between  the  teeth 
of  a  pair  of  mating  helical  gears,  the  angles  should  be  so  propor- 
tioned that,  in  the  diagrams,  line  DD'  will  be  approximately  at 
right  angles  to  ratio  line  AE.  On  the  other  hand,  to  give  the 
least  end  thrust  against  the  bearings,  line  DDf  should  make  an 
angle  of  45  degrees  with  the  margin  lines  AG  and  AG ',  in  the 
case  of  gears  with  axes  at  an  angle  of  90  degrees,  as  are  the  ones 
being  considered.  The  first  example,  worked  out  in  detail,  was 
solved  in  accordance  with  "good  practice,"  and  line  DD'  was 
located  about  one-half  way  between  the  two  positions  just  de- 
scribed, thus  giving  in  some  measure  the  advantage  of  a  com- 
parative absence  of  sliding  friction,  combined  with  as  small 
degree  of  end  thrust  as  is  practicable.  To  illustrate  some  of  the 
peculiarities  of  the  problem,  Case  3  will  now  be  solved  to  give 


i8 


SPIRAL  GEARING 


the  minimum  amount  of  sliding  friction,  neglecting  entirely  the 
end  thrust,  which  is  considered  to  be  taken  up  by  ball  thrust 
bearings  or  some  equally  efficient  device. 

By  trial  it  will  be  found  that,  with  the  same  number  of  teeth 
in  the  gear  and  with  the  same  pitch  as  in  Case  2,  giving,  in  Fig.  8, 
BO,  the  equivalent  diameter  of  gear  a,  a  value  of  i|  inch,  and  B'O, 
the  equivalent  diameter  of  gear  b,  a  value  of  3  inches,  as  in  Fig.  7, 
line  DD',  which  is  equal  to  twice  the  center  distance,  or  8  inches, 
can  then  lie  at  an  angle  of  about  90  degrees  with  AE,  thus  meet- 
ing the  condition  required  as  to  sliding  friction.  Thus  this  dia- 
gram, while  relating  to  gears  having  the  same  pitch  and  number 


Fig.  8.     Solution  of  Problem  No.  3  for  Minimum  Sliding  Friction 

of  teeth  as  Fig.  7,  has  an  entirely  different  appearance,  and 
gives  different  tooth  angles  and  center  distances,  solving  the 
problem  as  it  does  for  the  least  sliding  friction  instead  of  for 
equal  diameters  of  gears. 

Measuring  the  diagram  as  accurately  as  may  be,  the  following 
results  are  obtained:  Tooth  angle  of  gear  a  =  BOD  =  28°; 
tooth  angle  of  gear  b  =  angle  B'OD'  =  90°  -  28°  =  62°.  This 
is  the  preliminary  solution.  After  accurately  working  it  out  by 
the  process  before  described,  we  have  as  a  final  solution,  tooth 
angle  of  gear  a  =  28°  28';  tooth  angle  of  gear  b  =  61°  32'. 
From  this  the  remaining  data  can  be  calculated. 

For  designers  who  are  skillful  enough  to  solve  such  problems 
as  these  graphically  without  reference  to  calculations,  the  dia- 


RULES  AND  FORMULAS  19 

gram  may  be  used  for  the  final  solution.  The  variation  between 
the  results  obtained  graphically  and  those  obtained  in  the  more 
accurate  mathematical  solution  is  a  measure  of  the  skill  of  the 
draftsman  as  a  graphical  mathematician.  The  method  is  simple 
enough  to  be  readily  copied  in  a  notebook  or  carried  in  the  head. 
If  the  graphical  method  is  to  be  used  entirely,  it  will  be  best 
not  to  trust  to  the  cross-section  paper,  which  may  not  be  accu- 
rately ruled;  instead  skeleton  diagrams  like  those  shown  in  Figs.  6, 
7  and  8  may  be  drawn.  For  rough  solutions,  however,  to  be 
afterward  mathematically  corrected,  as  in  the  examples  con- 
sidered in  this  chapter,  good  cross-section  paper  is  accurate 
enough.  It  permits  of  solving  a  problem  without  drawing  a 
line.  Point  O  may  be  located  by  reading  the  graduations;  a 
pin  inserted  here  may  be  used  as  a  stop  for  the  rule,  from  which 
the  diameter  and  center  distance  are  read  directly;  dividing  AD, 
read  from  the  paper,  by  DD',  read  from  the  rule,  will  give  the 
sine  of  the  tooth  angle  of  the  gear  a. 

Basic  Rules  and  Formulas  for  Spiral  Gears.  —  The  rules  and 
formulas  given  in  the  foregoing  may  be  tabulated  as  shown  on 
the  following  page.  In  the  formulas  in  the  table  "Basic  Rules 
and  Formulas  for  Spiral  Gear  Calculations"  the  following  nota- 
tion is  used: 

Pn  =  normal  diametral  pitch  (pitch  of  cutter) ; 

D  =  pitch  diameter; 

N  =  number  of  teeth; 

a  =  spiral  angle; 

7  =  center  angle,  or  angle  between  shafts; 

C  =  center  distance; 
N'  =  number  of  teeth  for  which  to  select  cutter; 

L  =  lead  of  tooth  helix; 

5  =  addendum; 
W  =  whole  depth  of  tooth; 
Tn  =  normal  tooth  thickness  at  pitch  line; 

0  =  outside  diameter. 

Examples  of  Spiral  Gear  Problems.  —  As  proficiency  in  solv- 
ing spiral  gear  problems  can  be  obtained  only  by  a  great  deal  of 
practice,  a  number  of  examples  will  be  given  in  the  following, 


20 


SPIRAL  GEARING 


Basic  Rules  and  Formulas  for  Spiral  Gear  Calculations4 


— 

.jki 

In  the  formulas  N,  a 
numbers  of  teeth,  spira 
|                 for  either  gear  or  pinio 
ai                                    '                  tions  Nn,  Nh,  otn,  ahf  etc 

etc.,  are  the 
tl  angle,  etc., 
n;    the  nota- 
.,  refer  to  the 
mion  or  gear, 
)f  gears  a  and 

6  — 

.  ^  3--.-  *lj|l|ii§§|ll|    teeth  or  angles  in  the  p 
H§§^^^$^^    respectively,  in  a  pnir  ' 

W    t?  ' 

No. 

To  Find 

Rule 

Formula 

I 

Relation  between 
Shaft     and     Tooth 
Angles. 

The  sum  of  the  tooth  an- 
gles of  a  pair  of  mating  heli- 
cal  gears   is   equal   to   the 
shaft  angle. 

7  =  cta  +  ctb 

2 

Pitch  Diameter. 

Divide     the     number     of 
teeth  by  the  product  of  the 
normal  pitch  and  the  cosine 
of  the  tooth  angle. 

D           N 

PnCOSa 

3 

Center  Distance. 

Add    together    the    pitch 
diameters  of  the  two  gears 
and  divide  by  2. 

r        Da  +  Db 

2 

4 

Checking  Calcu- 
lations in  (2)  and 
(3). 

To  prove  the  calculations 
for  pitch  diameters  and  cen- 
ter  distance,    multiply   the 
number  of  teeth  in  the  first 
gear  by  the  tangent  of  the 
tooth  angle  of  that  gear,  and 
add  the  number  of  teeth  in 
the  second  gear  to  the  prod- 
uct;   the  sum  should  equal 
twice  the  product  of  the  cen- 
ter distance  multiplied  by 
the  normal  diametral  pitch, 
multiplied  by  the  sine  of  the 
tooth  angle  of  the  first  gear. 

Nb  +  (Na  X 
tanaa)  = 
2  CPnXsinaa 

5 

Number  of  Teeth 
for  which  to  Select 
Cutter. 

Divide  the  number  of  teeth 
in  the  gear  by  the  cube  of  the 
cosine  of  the  tooth  angle. 

N 

(cos  a)3 

6 

Lead    of    Tooth 
Helix. 

Multiply  the  pitch  diam- 
eter by  3.1416  times  the  co- 
tangent of  the  tooth  angle. 

L  =  7rI>Xcota 

7 

Addendum. 

Divide   i   by  the  normal 
diametral  pitch. 

s-  T 

*~  Pn 

8 

Whole  Depth  of 
Tooth. 

Divide  2.157  by  the  nor- 
mal diametral  pitch. 

w-   2'157 

T'~      Pn 

9 

Normal  Tooth 
Thickness  at  Pitch 
Line. 

Divide  1.571  by  the  nor- 
mal diametral  pitch. 

r           I-S7I 
In  =  -p— 
-t  n 

10 

Outside  Diam- 
etet. 

Add  twice  the  addendum 
to  the  pitch  diameter. 

0  =  D  +  2S 

From  MACHINERY'S  HANDBOOK. 


RULES  AND   FORMULAS 


21 


which  can  be  solved  by  simple  modifications  of  the  methods 
outlined  for  problems  of  Class  2.  The  same  reference  letters 
are  used  as  before. 

Example  i. —  Find  the  essential  dimensions  for  a  pair  of 
spiral  gears,  velocity  ratio  3  to  i,  center  distance  between  shafts 
5^  inches,  angle  between  shafts  38  degrees. 

First  obtain  a  preliminary  solution  by  the  diagram  shown  in 
Fig.  9.  Draw  lines  AG  and  AG\  making  an  angle  7  with  each 
other  equal  to  38  degrees,  the  angle  between  the  axes.  Locate 
the  ratio  line  AE  by  finding  any  point  such  as  Oi  between  AG 


Machinery 


Fig.  9.     Diagram  Applying  to  the  Solution  of  Example  i 

and  AGi,  that  is  distant  from  each  of  them  in  the  same  ratio  as 
that  desired  for  the  gearing.  In  the  case  shown,  it  is  6  inches 
from  AG\  and  2  inches  from  AG,  which  is  in  the  ratio  of  3  to  i 
as  required.  Through  Oi  draw  line  AE  which  may  be  called 
the  ratio  line.  Select  a  trial  number  of  teeth  and  pitch  of  cutter 
for  the  two  gears,  such,  for  instance,  as  36  teeth  for  the  gear 
and  12  for  the  pinion,  and  with  5  diametral  pitch  of  the  cutter. 
The  diameter  of  a  spur  gear  of  the  same  pitch  and  number  of 
teeth  as  the  gear  would  be  36  -f-  5  =  7.2  inches.  Find  the 
point  0  on  AE,  which  is  7.2  inches  from  AGi.  This  point  will 
be  2.4  inches  from  A G,  if  A E  is  drawn  correctly. 

Now  apply  a  scale  to  the  diagram,  with  the  edge  passing 


22  SPIRAL  GEARING 

through  0  and  with  the  zero  mark  on  line  AG,  shifting  it  to  differ- 
ent positions  until  one  is  found  in  which  the  distance  across  from 
one  line  to  another  (DDi  in  the  figure)  is  equal  to  twice  the 
center  distance,  or  10.25  inches.  If  a  position  of  the  rule  cannot 
be  found  which  will  give  this  distance  between  lines  AG  and  AG\, 
new  assumptions  as  to  number  of  teeth  and  diametral  pitch  of 
the  gear  and  pinion  must  be  made  which  will  bring  point  0  in  a 
location  where  line  DDi  may  be  properly  laid  out.  DD{  being 
drawn,  the  problem  is  solved  graphically.  The  tooth  angle  of 
the  gear  is  BiODi,  or  ab,  while  that  of  the  pinion  is  BOD,  or  aa. 
ODi  will  be  the  pitch  diameter  of  the  gear,  and  OD  the  pitch 
diameter  of  the  pinion. 

To  obtain  the  dimensions  more  accurately  than  can  be  done 
by  the  graphical  process,  the  pitch  diameters  should  be  figured 
from  the  tooth  angles  we  have  just  found.  To  do  this,  divide 
the  dimensions  OBi  and  OB  for  gear  and  pinion,  by  the  cosine 
of  the  tooth  angles  found  for  them.  If  they  measure  on  the 
diagram,  for  instance,  21  degrees  50  minutes  and  16  degrees 
10  minutes  respectively  (note  that  the  sum  of  aa  and  ab  must 
equal  7),  the  calculation  will  be  as  follows: 

7.2  +  0.92827  =    7.7563  =  Db 

2.4-7-  0.96046  =   2.4988  =  Da 
IO.255I  =  2  C 

The  value  we  thus  get,  10.2551  inches,  for  twice  the  center 
distance,  is  somewhat  larger  than  the  required  value,  10.250 
inches.  We  have  now  to  assume  other  values  for  aa  and  e*6, 
until  we  find  those  which  give  pitch  diameters  whose  sum  equals 
twice  the  center  distance.  Assume,  for  instance,  that  ab  =  21 
degrees  43  minutes,  then  aa  =  38  degrees  —  21  degrees  43  min- 
utes =  1 6  degrees  17  minutes.  We  now  have: 

7.2  -f-  0.92902  =    7.7501  =  Db 

2.4  -j-  0.95989  =    2.5003  =  Dg 

IO.25O4  =  2  C 

This  value  for  twice  the  center  distance  is  so  near  that  required 
that  we  may  consider  the  problem  as  solved.  The  other  dimen- 
sions for  the  outside  diameter,  lead,  etc.,  may  be  obtained  as  for 


RULES  AND  FORMULAS 


23 


spiral  gears  at  right  angles,  and  as  described  in  the  previous  part 
of  this  chapter. 

Example  2.  —  Find  the  essential  dimensions  of  a  pair  of  spiral 
gears,  velocity  ratio  8  to  3,  center  distance  between  shafts  9^6 
inches,  angle  between  shafts  40  degrees. 

The  diagram  for  solving  this  problem  is  shown  in  Fig.  10. 
The  axis  lines  AG\  and  AG  are  drawn  as  before  and  the  ratio 
line  AE  is  drawn  in  the  ratio  of  8  to  3,  or  1 6  to  6,  by  the  same 
method  as  just  described.  A  point  0  is  found  having  a  location 
corresponding  to  64  teeth  and  5  pitch  for  the  gear,  and  24  teeth 
for  the  pinion.  This  gives  distance  OBi  =  1 2.8  inches,  and  OB  = 


Machinery 


Fig.  10.     Diagram  Applying  to  the  Solution  of  Example  2 

4.8  inches,  by  which  position  0  is  so  located  that  a  line 
can  be  drawn  through  it  at  a  convenient  angle,  and  with  a  length 
equal  to  twice  the  center  distance,  or  18.625  inches.    We  measure 
the  angle  for  a  preliminary  graphical  solution  as  before,  and  then 
by  trial  find  the  final  solution,  in  which  angle  ab  is  17  degrees 
45  minutes,  and  aa  is  22  degrees  15  minutes  as  follows: 
12.8  -^  0.95240  =  13.4397  =  A 
4.8  -f-  0.92554  =    5.1862  =  Dg 

18.6259  =  2  C 

This  gives  the  value  of  twice  the  center  distance  near  enough 
for  gears  of  this  size. 


SPIRAL  GEARING 


Example  3.  —  Find  the  essential  dimensions  for  a  pair  of  spiral 
gears,  velocity  ratio  5  to  2,  center  distance  between  shafts  4I1F 
inches,  angle  of  shafts  18  degrees. 

The  diagram  for  solving  this  problem  is  shown  in  Fig.  1 1 .  The 
axis  lines  AGi  and  AG  are  drawn  as  before,  and  the  ratio  line  AE 
is  drawn  in  the  ratio  of  5  to  2,  by  the  same  method  as  just  de- 
scribed. A  point  O  is  found  having  a  location  corresponding 
to  45  teeth  and  8  pitch  for  the  gear,  and  18  teeth  for  the  pinion. 
This  gives  distance  OBi  =  5.625  inches,  and  OB  =  2.250  inches, 
in  which  position  0  is  so  located  that  line  DDi  can  be  drawn 
through  it  at  a  convenient  angle,  and  with  a  length  equal  to 
twice  the  center  distance,  or  8.125  inches.  We  measure  the 
angles  for  a  preliminary  mathematical  solution  as  before,  and 


Fig.  ii.     Diagram  Applying  to  the  Solution  of  Example  3 

then   by    trial   find   the   final   solution,  in   which  angle  ab  is 
1 6  degrees  45  minutes  and  aa  is  i  degree  15  minutes  as  follows: 
5.625  -s-  0.95757  =  -5.8742  =  Db 
2.250  -T-  0.99976  =  2.2505  =  Da 
8.1247  =  2C 

It  is  often  a  matter  of  great  difficulty,  when  the  center  angle  y 
is  as  small  as  in  this  case,  to  find  a  location  for  point  O  such  that 
standard  cutters  can  be  used,  and  that  line  DDi  can  be  drawn  of 
the  proper  length  through  0  without  bringing  D  to  the  left  of 
B,  or  DI  to  the  left  of  BI.  It  will  be  noticed  in  this  case  that  to 
make  the  center  distance  come  right,  angle  aa  had  to  be  made 
very  small,  so  that  the  pinion  is  practically  a  spur  gear.  In 
some  cases,  to  get  the  proper  center  distance,  it  may  be  neces- 
sary to  so  draw  line  DDi  that  one  of  the  tooth  angles  is  measured 


RULES,  AND  FORMULAS  25 

on  the  left  side  of  BO  or  B±O.  Such  a  case,  for  instance,  is  shown 
in  the  position  of  df)d.  When  a  line  has  to  be  drawn  like  this, 
the  tooth  angles  aaf  and  abf  are  opposite  in  inclination,  instead 
of  having  them,  as  usual,  either  both  right-hand  or  both  left- 
hand.  In  Fig.  12  are  shown  gears  drawn  in  accordance  with  the 
location  of  line  DDi  of  Fig.  n,  while  Fig.  13  shows  a  pair  drawn 
in  accordance  with  ddi  of  the  same  diagram,  which  will  illustrate 
the  state  of  affairs  met  with  in  cases  of  this  kind.  This  expedient 
of  making  one  spiral  gear  right-hand  and  one  left-hand  should 
never  be  resorted  to  except  in  case  of  extreme  necessity,  as  the 
construction  involves  a  very  wasteful  amount  of  friction  from 
the  sliding  of  the  teeth  on  each  other  as  the  gears  revolve. 


Machinery 


Figs.  12  and  13. 


Comparison  between  Two  Pairs  of  Gears  determined 
from  the  Diagram  in  Fig.  n 


Demonstration  of  Grant's  Formula.  —  As  already  mentioned, 
the  number  of  teeth  for  which  the  cutter  should  be  selected  for 
cutting  a  helical  gear,  is  found  from  the  formula 


in  which  Nf  =  number  of  teeth  for  which  cutter  is  selected; 
N  =  actual  number  of  teeth  in  helical  gear; 
a  =  angle  of  tooth  with  axis. 

Note  that  cos3  a  is  equivalent  to  (cos  a)3. 

A  demonstration  of  this  formula  was  presented  by  Mr.  H.  W. 


26 


SPIRAL  GEARING 


Henes  in  MACHINERY,  April,  1908.  This  demonstration  is  as 
follows: 

Let  Pn  be  the  perpendicular  distance  between  two  consecutive 
teeth  on  the  spiral  gear,  and  let  A  be  the  diameter  of  the  spiral 
gear.  Let  the  gear  be  represented  as  in  Fig.  14,  and  pass  a  plane 
through  it  perpendicular  to  the  direction  of  the  teeth.  The 
section  will  be  an  ellipse  as  shown  in  CEDF.  Designate  the 
semi-major  and  semi-minor  axes  by  a  and  &,  respectively. 

Now  Nr  is  the  number  of  teeth  which  a  spur  gear  would  have 


Machinery 


Fig.  14.     Diagram  for  Deriving  Formula  for  Determining  Spur  Gear 
Cutter  to  be  used  for  Cutting  Spiral  Gears 

if  its  radius  were  equal  to  the  radius  of  curvature  of  the  ellipse 
at  E.    Therefore,  it  is  required  to  determine  the  radius  of  this 
curvature  of  the  ellipse.     This  is  done  as  follows: 
From  the  figure  we  have: 

2&  =  axis£F  =  A  (i) 

•      /^T-»  t-iT-r  ///  Dl  /     \ 

2  a  =  axis  CD  =  GH  = =  — —  (2) 

cos  a      cos  a 

From  (i)  and  (2)  we  have  for  a  and  b, 


(3) 


RULES  AND  FORMULAS  27 

A 

a  =  -  —  (4) 

2  cos  a 

It  is  known,  and  shown  by  the  methods  of  calculus,  that  the 
minimum  curvature  of  an  ellipse,  that  is,  the  curvature  at  E 

or  F,  equals  —  .     Taking  the  values  of  a  and  b  found  in  (3)  and 
(4),  we  have  the  curvature  at  E: 


~  b  2  4  DI  COS2  a.         2  COS2  a  ,  N 

Curvature  =  -,  =  -^-  -    L__         __  (s) 

4  COS2  a 

It  is  also  shown  in  calculus  that  the  curvature  is  equal  to  — 

K. 

where  R  is  the  radius  of  curvature  at  the  point  E.    Therefore 
from  (5)  we  have  : 

i      2  cos2  a.         ,  ,  ,         „          DI  x,x 

-  =  —  -  -    and  thus  R  =  -  ~-  (6) 

R  DI  2  COS2  a 

Formula  (6)  can  also  be  arrived  at  directly,  without  reference 
to  the  minimum  curvature  of  the  ellipse,  by  introducing  the 
formula  for  the  radius  of  curvature  in  the  first  place.  The 
curvature  is  simply  the  reciprocal  value  of  the  radius  of  curva- 
ture, and  is  only  a  comparative  means  of  measurement.  The 
radius  of  curvature  of  an  ellipse  at  the  end  of  its  short  axis  is 

a2 

—  >  from  which  Formula  (6)  may  be  derived  directly  by  intro- 

0 

ducing  the  values  of  a  and  b  from  Equations  (3)  and  (4). 

Having  now  found  the  radius  of  curvature  of  the  ellipse  at  E, 
we  proceed  to  find  the  number  of  teeth  which  a  spur  gear  of  that 
radius  would  have.  From  Fig.  14  we  have: 

AB  =    *Z-  (7) 

COS  a 

Now,  if  AB  be  multiplied  by  the  number  of  teeth  of  the  spiral 
gear,  we  shall  obtain  a  quantity  equal  to  the  circumference  of 

the  gear;   that  is: 

p 

AB  X  N  =  irDi,   and  since  AB  =  —  —  from  (7) 

COS  a 

-^=-  X  N  =  TrA  (8) 

COS  a 


28  SPIRAL  GEARING 

Since  N'  is  the  number  of  teeth  which  a  spur  gear  of  radius  R 
would  have,  then, 

N'=^f  (9) 

-Ln 

In  Equation  (9)  the  numerator  of  the  fraction  is  the  circum- 
ference of  the  spur  gear  whose  radius  is  R,  and  the  denominator 
is  the  circular  pitch  corresponding  to  the  cutter. 

From  Equation  (6)  we  have: 


2  COS2  a 
Substituting  this  value  of  R  in  (9),  we  have: 


From  Equation  (8)  we  have: 
A  =  - 

Tr 
Substitute  this  value  of  D\  in  Equation  (10)  and  we  have: 


A  =  --  (ii) 

Trcosa 


2  PnTT  COS3  a 

or 


COS  a: 


Since  N  is  the  number  of  teeth  in  a  spiral  gear  and  N'  is  the 
number  of  teeth  in  a  spur  gear  which  has  the  same  radius  as  the 
radius  of  curvature  of  the  ellipse  referred  to,  this  is  the  equiv- 
alent of  saying  that  the  cutter  to  be  used  should  be  correct 
for  a  number  of  teeth  which  can  be  obtained  by  dividing  the 
actual  number  of  teeth  in  the  gear  by  the  cube  of  the  cosine 
of  the  tooth  angle.  Since  the  cosine  of  angle  is  always  less  than 
unity,  its  cube  will  be  'still  less,  so  N'  is  certain  to  be  greater 
than  N,  which  will  account  for  the  fact  that  spiral  gears  of  less 
than  12  teeth  can  be  cut  with  the  standard  cutters. 


CHAPTER  II 

FORMULAS  FOR  SPECIAL  CASES   OF   SPIRAL  GEAR 

DESIGN 

THE  rules  and  formulas  given  in  the  tabulated  arrangement  in 
the  preceding  chapter  are  presented  in  the  same  order  as  they 
would  ordinarily  be  used  by  the  designer  when  calculating  a 
pair  of  spiral  gears.  The  formulas,  however,  cannot  be  directly 
applied  to  all  cases  of  spiral  gear  problems,  except  by  the  use  of 
a  graphical  method,  as  outlined,  and  a  complete  set  of  formulas 
for  each  of  the  sixteen  different  cases  which  are  most  frequently 
met  with  is,  therefore,  given  in  the  following,  together  with  an 
example  for  each  case.  These  sixteen  cases  are: 

1.  Shafts  parallel,  ratio  equal  and  center  distance  approxi- 

mate. 

2.  Shafts  parallel,  ratio  equal  and  center  distance  exact. 

3.  Shafts  parallel,  ratio  unequal  and  center  distance  approxi- 

mate. 

4.  Shafts  parallel,  ratio  unequal  and  center  distance  exact. 

5.  Shafts  at  right  angles,  ratio  equal  and  center  distance 

approximate. 

6.  Shafts  at  right  angles,  ratio  equal  and  center  distance 

exact. 

7.  Shafts  at  right  angles,  ratio  unequal  and  center  distance 

approximate. 

8.  Shafts  at  right  angles,  ratio  unequal  and  center  distance 

exact. 

9.  Shafts  at  45-degree  angle,  ratio  equal  and  center  distance 

approximate. 

10.  Shafts  at  45-degree  angle,  ratio  equal  and  center  distance 

exact. 

11.  Shafts  at  45-degree  angle,  ratio  unequal  and  center  dis- 

tance approximate. 

29 


SPIRAL  GEARING 


12.  Shafts  at  45-degree  angle,  ratio  unequal  and  center  dis- 

tance exact. 

13.  Shafts  at  any  angle,  ratio  equal  and  center  distance  ap- 

proximate. 

14.  Shafts  at  any  angle,  ratio  equal  and  center  distance  exact. 

15.  Shafts  at  any  angle,  ratio  unequal  and  center  distance 

approximate. 

16.  Shafts  at  any  angle,  ratio  unequal  and  center  distance  exact. 


Fig.  9 


Fig.  1O 


Fig.  1 1 


Fig.  12 


Figs,  i  to  12.  Thrust  Diagrams  for  Spiral  Gears — Direction  of  Thrust 
depends  upon  Direction  of  Rotation,  Relative  Position  of  Driver  and 
Driven  Gear,  and  Direction  of  Spiral 

The  proofs  of  the  more  complicated  formulas  are  given  in  the 
explanatory  matter  preceding  each  specific  set  of  formulas.  All 
the  information  necessary  for  the  shop  operations  is  given,  in- 
cluding the  number  of  teeth  marked  on  the  spur  gear  cutter 
used,  and  the  lead  of  the  spiral,  which  data  are  often  omitted  on 
the  drawing,  but  which  always  ought  to  be  given.  If  omitted, 
the  operator  of  the  milling  machine  or  gear  cutter  must  deter- 
mine these  data  himself,  and  this  is  not  a  commendable  method. 


RULES  AND  FORMULAS 


31 


Procedure  in  Calculating  Spiral  Gears.  —  One  of  the  first 
steps  necessary  in  spiral  gear  design  is  to  determine  the  direction 
of  the  thrust,  if  the  thrust  is  to  be  taken  in  one  direction  only. 
When  the  direction  of  the  thrust  has  been  determined  and  the 
relative  position  of  the  driver  and  driven  gear  is  known,  the 
direction  of  spiral  (right-  or  left-hand)  may  be  found.  The 
thrust  diagrams,  Figs,  i  to  28,  are  used  for  finding  the  direction 
of  spiral.  The  arrows  at  the  end  bearings  of  the  gears  indicate 


Flg.,28 


Figs.  13  to  28.  Thrust  Diagrams  for  Spiral  Gears  —  Direction  of  Thrust 
depends  upon  Direction  of  Rotation,  Relative  Position  of  Driver  and 
Driven  Gear,  and  Direction  of  Spiral 

the  direction  of  the  reaction  against  the  thrust  caused  by  the 
tooth  pressure.  The  direction  of  the  thrust  depends  on  the 
direction  of  spiral,  the  relative  positions  of  driver  and  driven 
gear  and  the  direction  of  rotation.  If  the  exact  condition  with 
regard  to  thrust  is  not  found  in  the  diagrams,  it  may  be  obtained 
by  changing  any  one  of  these  three  conditions;  that  is,  in  Fig.  i 
the  thrust  may  be  changed  to  the  opposite  direction  by  inter- 
changing driver  or  driven  gear,  by  reversing  the  direction  of 


32 


SPIRAL  GEARING 


rotation  or  by  changing  the  direction  of  spiral.  Any  one  of 
these  alterations  will  produce  a  thrust  in  the  opposite  direc- 
tion. 

The  conditions  of  design  will  determine  the  nature  of  center 
distances,  whether  they  must  be  exact  or  approximate.  The 
strength  of  tooth  needed,  or  sometimes  the  cutters  on  hand,  will 
determine  the  normal  pitch  of  the  gear.  The  formulas  given 
for  the  different  conditions  of  spiral  gearing  are  all  based  on  the 
normal  diametral  pitch  which  is  the  same  as  the  diametral  pitch 
of  the  cutter  used.  The  number  of  teeth  in  each  gear  is,  of 


LEAD  = 


Machinery,  N.  Y. 


Figs.  29  and  30.     Diagrams  for  Derivation  of  Formulas 

course,  determined  by  the  required  speed  ratio  of  the  shafts. 
The  angle  of  spiral  depends  on  the  conditions  of  the  design,  and 
the  relative  position  of  the  shafts.  If  the  shafts  are  parallel, 
the  gears  may  be  of  the  herringbone  type,  when  an  angle  as 
great  as  45  degrees  may  be  used,  as  there  is  no  end  thrust.  When 
used  as  shown  in  the  thrust  diagrams,  the  spiral  angle  should  not 
exceed  20  degrees  with  parallel  shafts,  thus  avoiding  excessive 
end  thrust.  In  order  to  obtain  smooth  running  gears,  the  spiral 
angle  should  also  be  such  that  one  end  of  the  tooth  remains  in 
contact  until  the  opposite  end  of  the  following  tooth  has  found  a 
bearing,  as  indicated  at  V  and  W  in  Fig.  30. 


RULES  AND  FORMULAS  33 

i.  Shafts  Parallel,  Ratio  Equal  and  Center  Distance  Approx- 
imate. —  vThis  case  is  met  with  in  new  designs,  where  an  exact 
center  distance  is  of  no  importance.  The  five  factors,  direction 
of  spiral,  approximate  center  distance,  normal  diametral  pitch, 
number  »of  teeth  and  angle  of  spiral  are  first  determined  upon. 
Then,  from  the  formulas  given,  the  required  data  are  found  as 
shown  in  the  example  given.  The  following  shows  the  derivation 
of  the  formulas;  in  Fig.  29,  let  a  be  the  angle  of  spiral  with  the 
axis  of  the  gear;  let  H  be  the  distance  from  one  tooth  to  the  next, 
measured  on  the  circumference  of  the  pitch  circle,  and  K  the 

jr 

normal  circular  pitch  of  the  gear.     Then  H  =  --     The  diam- 

cos  a 

etral  pitch  =  —   -^  —  —  -  -    Let  Pn  =  normal  diametral  pitch, 
circular  pitch 

or  diametral  pitch  of  cutter  used.     Then  Pn  ='  -••>  or  transposing, 

K. 

K  —  ~—  -     If  N  =  number  of  teeth  in  gear,  the  circumference 

*  n 

NH 

of  the  pitch  circle  =  N  X  H,  and  -  =  pitch  diameter  =  D. 

Hence, 


TT        cos  a       TTCOSCK       Pn 

In  all  cases  where  the  shafts  are  parallel,  the  value  of  a  is  the 
same  for  both  gears.  The  outside  diameter  of  the  spiral  gear  is 

found  exactly  as  in  spur  gears,  by  adding  —  to  the  pitch  diam- 

•*  n 

N 

eter.     The  derivation  of  formula  T  =  —  r—  was  treated  fully 

cos3  a 

in  the  preceding  chapter. 

A  standard  spur  gear  cutter  is,  of  course,  used  in  cutting  spiral 
gears,  but  Ty  the  number  of  teeth  marked  on  it,  will  probably 
not  be  the  actual  number  of  teeth  in  the  spiral  gear  to  be  cut, 
T  depending,  as  the  formula  shows,  on  the  spiral  angle.  As 
to  the  lead  of  spiral,  let  Fig.  30  represent  a  spiral  gear,  where 
the  oblique  line  is  the  path  of  the  tooth  unfolded.  Then  L  = 
irD  cot  a,  in  which  L  =  lead  of  spiral,  irD  =  pitch  circumference 
and  a  =  spiral  angle. 


34 


SPIRAL  GEARING 


To  find: 
i.  D 


Formulas,  Case  i 
Given  or  assumed: 

i.  Hand  of  spiral  on  driver  or  driven  gear  de- 
pending on  rotation  and  direction  in  which 
thrust  is  to  be  received.  • 

Ca  =  approximate  center  distance. 
Pn  =  normal  pitch  (pitch  of  cutter). 
N  =  number  of  teeth. 
a  =  angle  of  spiral  (usually  less  than  20 
degrees  to  avoid  excessive  end  thrust). 


pitch  diameter  = 


N 


Pn cos  a 


2.  O  =  outside  diameter  =  D  +  •— - 

Pn 

3.  T  =  number  of  teeth  marked  on  cutter  = 

4.  L  =  lead  of  spiral  =  irD  cot  a. 

Example 


N 


cos3 


a 


Given  or  assumed: 
i.   See  illustration. 

3-    Pn  =  8. 

5.    a  =  15  degrees. 

To  find: 

N  24 

Pn  cos  a      8  X  0.9659 

2 


2.   Ca  =  3  inches. 
4.    N  =  24. 


i.       - 


3.106  inches. 


2. 


O  =  3.106  +  o  =  3-356  inches. 

o 


3.    T  = 


=  —  =  26.6,  say  27  teeth. 
0.9  \ 


4.    L  =  7rZ)coti5°  =  3.1416X3.106X3.732=36.416  inches. 

2.  Shafts  Parallel,  Ratio  Equal  and  Center  Distance  Exact.  — 
The  spiral  angle  a  is  found  in  terms  of  the  number  of  teeth  in 
each  gear,  normal  diametral  pitch  and  pitch  diameter,  which 
latter  is,  of  course,  equal  to  the  center  distance. 

N  N 

From  Formula  (i)  in  Case  (i),D  =  -—  —  —  ,  or  cos  a  = 


Pn cos  a 
The  remaining  formulas  are  found  as  in  the  first  case. 


PnXD 


RULES  AND  FORMULAS 


35 


Formulas,  Case  2 
Given  or  assumed: 

1.  Position  of  gear  having  right-  or  left-hand 

spiral,  depending  on  rotation  and  direc- 
tion in  which  thrust  is  to  be  received. 

2.  C  —  exact  center  distance  =  pitch  diam- 

eter D. 

3.  Pn  =  normal  pitch  (pitch  of  cutter). 

4.  N  =  number  of  teeth  in  each  gear. 


To  find: 


I.   COS  a. 


N 

PnD 


N 


2.         O  =  outside  diameter  =  D  +  — - 

f' n 

3!         T  =  number  of  teeth  marked  on  cutter  = 

4.         L  =  lead  of  spiral  =  irD  cot  a. 

a  is  usually  less  than  20  degrees  to  avoid  excessive  end  thrust. 


cosd 


Given  or  assumed: 
i.   See  illustration. 
3-   Pn  =  8. 

To  find: 

N 


Example 

2.     C 


3  inches. 
22. 


i.   cos  a 


PnD 


0.9166,  or  a  =  23°  34'. 


2. 


O 


-^  =  3  +  o  =  3i  inches. 


T  = 


N 


22 


28.2,  say  28  teeth. 


cos3  a      (o.92)3 

4.         L  =  irDcota  =  3.1416  X  3  X  2.29  =  21.58  inches. 

3.  Shafts  Parallel,  Ratio  Unequal  and  Center  Distance 
Approximate.  —  The  formulas  for  this  case  are  practically  the 
same  as  for  Case  (i),  and  are  derived  in  the  same  manner.  The 
spiral  angle  is  of  the  same  value  in  both  gears,  as  in  all  spiral 
gears  with  parallel  shafts,  but,  of  course,  of  a  different  direction 
(hand)  in  each  gear. 


SPIRAL  GEARING 


Formulas,  Case  3 
Given  or  assumed: 

1.  Position   of   gear  having   right-   or   left-hand 

spiral,  depending  upon  rotation  and  direction 
in  which  thrust  is  to  be  received. 

2.  Ca  =  approximate  center  distance. 

3.  Pn  =  normal  pitch. 

4.  N  =  number  of  teeth  in  large  gear. 

5.  n  =  number  of  teeth  in  small  gear. 

6.  a  =  angle  of  spiral. 


To  find: 

1.  D  =  pitch  diameter  of  large  gear  = 

2.  d  —  pitch  diameter  of  small  gear  = 


N 


Pn  cos  a 


3.    0  =  outside  diameter  of  large  gear  =  D  -\-  -—- 


4. 
5. 


o  —  outside  diameter  of  small  gear 


+ 


T  =  number    of    teeth  marked   on   cutter    (large   gear) 

N 

COS3  a 

6.  /  =  number  of  teeth  marked  on  cutter  (small  gear)  =  - 

cos3  a 

7.  L  =  lead  of  spiral  on  large  gear  =  irD  cot  a. 

8.  /  =  lead  of  spiral  on  small  gear  =  ird  cot  a. 

9.  C  =  center  distance  (if  not  right  vary  «)  =  %  (D  +  d). 

Example 
Given  or  assumed: 

i.   See  illustration.    2.   Ca  =  17  inches.      3.  Pn  =  2. 

4.   N  —  48.  5.     n  =  20.  6.     a  =  20  degrees. 

To  find: 

N  48 


i. 


COS  a   2  X  0.9397 


2.   d  = 


20 


Pn  COS  CK    2  X  0.9397 


=  25.541  inches. 
=  10.642  inches. 


_2_ 
P* 


25.541  +  ~  =  26.541  inches. 


RULES  AND  FORMULAS 


37 


*J  *) 

o  •=  d  +  —  =  10.642  +  -  =  11.642  inches. 

*•  2 


5- 

6. 


T  = 


t  = 


N 


cos3 


a 


^=57-8,  say  58  teeth. 


2O 


=  24.1,  say  24  teeth. 


cos3  a.      (0.9397)* 

7.  Z,  =  xDcota  =  3.1416  X  25.541  X  2.747  =  220.42  inches. 

8.  /  =  Trdcotc*  =  3.1416  X  10.642  X  2.747  =  91.84  inches. 
9-    C  =  %  (D  -}-  d)  =  %  (25.541  +  10.642)  =  18.091  inches. 

4.   Shafts  Parallel,  Ratio  Unequal  and  Center  Distance  Ex- 
act. —  In  this  case  the  sum  of  the  two  pitch  diameters  of  the 

N  n 


gears,  or  twice  the  center  distance  is 


Pn  COS  a.       Pn  COS  a. 


=  2C 


from  which  cos  a  = 


N  +  n 


2PnC 

N  and  n  are  the  numbers  of  teeth  in  the  respective  gears,  and 
C  the  center  distance.  The  remaining  eight  formulas  are  simi- 
lar to  those  of  the  other  cases. 


L 


Formulas,  Case  4 

Given  or  assumed: 

1.  Position   of  gear  having  right-  or  left-hand 

spiral,  depending  upon  rotation  and  direc- 
tion in  which  thrust  is  to  be  received. 

2.  C  =  exact  center  distance. 

3.  Pn  =  normal  pitch  (pitch  of  cutter). 

4.  N  =  number  of  teeth  in  large  gear. 

5.  n  =  number  of  teeth  in  small  gear. 


To  find: 


I.    COS  a  = 


2.         D  =  pitch  diameter  of  large  gear  = 


N 


d  =  pitch  diameter  of  small  gear  = 


Pn cos  a 

n 
Pn COS  a 


O  =  outside  diameter  of  large  gear  =  D 


38  SPIRAL  GEARING 

5.  o  =  outside  diameter  of  small  gear  =  d  H 

*  • 

6.  T  —  number  of  teeth  marked  on  cutter  (large  gear) 

N 

COS3  a 

7.  /  =  number  of  teeth  marked  on  cutter  (small  gear) 

n 


8.  L  =  lead  of  spiral  (large  gear)  =  irD  cot  a. 

9.  /   =  lead  of  spiral  (small  gear)  =  ird  cot  a. 

Example 
Given  or  assumed: 

i.  See  illustration.     2.   C  =  18.75  inches.     3.  Pn  =  4. 
4.  N  =  96.  5.   n  =  48. 

To  find: 

N  +  n          96  +  48  ,0    ,, 

1.  cos  a  =  — =—^  = rt- —  =  0.96.  or  a  —  16    16  . 

2PnC        2X4X18.75 

A7  96  .    , 

2.  Z)  =  — =  — 2 =  25  inches. 

Pn  cos  a      4  X  0.96 

n  48  .     , 

i.          a  =  — =  — •  =  12.5  inches. 

Pn  cos  a      4  X  0.96 

2  2 

4.  0=Z>-f~=25+-  =  25.5  inches. 

-Tn  4 

5.  o  =  d  +  -^~=  12.5  -f  -  =  13  inches. 

rn  4 

6.  r--4-  =  r4^=I°8  teeth. 

cos3  a.       (o.96)3 

7.  t  =  —^-  =  .  4,.3  =  54  teeth. 

cos3  a      (o.96)3 

8.  L  =  irDcota  =  3.1416  X  25     X  3.427  =  269.15  inches. 

9.  /  =  irdcota  =  3.1416  X  12.5  X  3.427  =  134-57  inches. 

5.  Shafts  at  Right  Angles,  Ratio  Equal,  and  Center  Distance 
Approximate.  —  The  sum  of  the  spiral  angles  of  both  gears  must 
in  this,  and  in  the  three  following  cases,  equal  90  degrees,  and 
the  direction  of  the  spiral  must  be  the  same  for  both  gears.  When 


RULES  AND  FORMULAS 


39 


the  spiral  angles  of  both  gears  equal  45  degrees,  the  pitch  diam- 
eters of  both  gears  will  be  equal,  and  are  found  by  the  formula: 


D 


Pncos45° 

The  other  formulas  are  the  same  as  those  given  in  the  previous 
cases. 

Formulas  ,  Case  5 

When  the  spiral  angles  are  45  degrees,  the  gears  are  exactly 
alike;  when  other  than  45  degrees,  the  sum  of 
the  spiral  angles  must  equal  90  degrees. 
Given  or  assumed: 

1.  Position  of  gear  having  right-  or  left-hand 

spiral,  depending  on  the  rotation  and 
direction  in  which  the  thrust  is  to  be 
received. 

2.  C0  =  approximate  center  distance. 

3.  Pn  =  normal  pitch  (pitch  of  cutter). 

4.  N  =  number  of  teeth. 

5.  a    =  angle  of  spiral. 

To  find: 

(a)  When  spiral  angles  are  45  degrees. 

N 

1.  D  =  pitch  diameter  =  -  :r 

0.707  1  1  Pn 
2 

2.  0  =  outside  diameter  =  D  +  — 

±n 

3.  T  =  number  of  teeth  marked  on  cutter 

4.  L  —  lead  of  spiral  =  irD. 

5.  C  =  center  distance  =  D. 


N 


°-353 


(b)  When  spiral  angles  are  other  than  45  degrees. 

N 

i.  D  =  pitch  diameter  = 


2.   T  =  number  of  teeth  marked  on  cutter  = 


N 

r 
cos3 


3.  C  —  center  distance  =  sum  of  pitch  radii. 

4.  L  =  lead  of  spiral  =  irD  cot  a. 


40  SPIRAL   GEARING 

Example 
Given  or  assumed: 

i.   See  illustration.        2.   Ca  =  2.5  inches.        3.  Pn  =  10. 

4.   N  =  1 8  teeth.  5.   a  =  45  degrees. 

To  find: 

1.  D  =  -         —Tr  =  • —  —  =  2.546  inches. 

0.707II  Pn         0.707II    X   10 

2  2 

2.  O  =  D  +  -£-  =  2.546  +  —  =  2.746  inches. 

Jtn  IO 

3.  T  =  — r-  =  —   -  =  51  teeth. 

cos3  a      0.353 

4.  L  =  irD  X  i  =  3.i4x6  X  2.546  =  7.999  inches. 

6.  Shafts  at  Right  Angles,  Ratio  Equal  and  Center  Distance 
Exact.  —  After  deciding  upon  an  approximate  spiral  angle  of  one 
gear,  the  number  of  teeth  in  each  gear  is  found  nearest  the  value 
CPn  cos  (f>,  where  <}>  is  this  approximate  spiral  angle,  and  C  the 

N 

exact    center    distance.     If  —    —  =  C.    or    N  =  CPn  cos  a. 

Pn  cos  a 

then  N  would  be  an  approximate  number  of  teeth  with  an  exact 
spiral  angle,  but  by  making  a.  =  $,  an  approximate  angle,  then 
N  =  CPn  cos  <£,  or  an  exact  number  of  teeth,  after  which,  as 
shown  in  the  following,  an  exact  angle  a  can  be  found,  for 

N  N 

Pn  COS  a         Pn  COS  j8 

where  a  is  the  exact  spiral  angle  of  one  gear,  and  /3  the  exact 
spiral  angle  of  the  other,  but  /3  =  90°  —  a,  or  cos  0  =  sin  a. 
Then, 


PnCOSa      Pn  since 
or 

I  I  2  CPn 


cos  a      sin  a          N 
Multiplying  by  sin  a  cos  a  gives: 


2  CPn 

sin  a.  +  cos  a  =  — 77-^  sin  a  cos  a. 


Squaring, 


sin2  a  -f  2  sin  a  cos  a  -f  COS2  a  =  ~ 


RULES  AND  FORMULAS 


But      • 

2  sin  a  cos  a  =  sin  2  a,  and  sin2  a  +  cos2  a  =  i. 
Further, 

sin2  a  cos2  a  =  |  sin2  2  a. 
Then 


I  +  sin  2  a  = 


Sin  2  a 


or 


sm  2  a  - 


2  a  = 


Solving  this  equation  we  get: 

N2      >rw 


c2Pn2  +  \2C2pn2/ 


sin  2  a  = 
which  is  the  equation  for  finding  twice  the  required  angle. 

Formulas,  Case  6 

Gears  have  same  direction  of  spiral  but  prob- 
ably different  pitch  diameters  and  spiral 
angles;  the  sum  of  the  latter  must  be  90 
degrees. 

Given  or  assumed: 

1.  Position  of  gear  having  right-  and  left- 

hand  spiral  depending  on  rotation 
and  direction  in  which  thrust  is  to  be 
received. 

2.  Pn  =  normal  pitch  (pitch  of  cutter). 

3.  0  =  approximate  spiral  angle  of  one  gear. 

4.  C  =  center  distance. 

5.  ]V  =  number  of  teeth  =  nearest  whole  number  to  CPn  X 

cos  0. 
To  find: 

i.     a  =  spiral  angle  of  one  gear. 
N2 


sin  2  a. 


C2P 


/JVM 

\2  C2P  2/ 


2.  /3  =  spiral  angle  of  other  gear  =  90°  —  a. 

N 

3.  D  =  pitch  diameter  of  one  gear  =  — 

Pn cos  a 


42  SPIRAL  GEARING 

4.  d  —  pitch  diameter  of  other  gear  = 

5.  O  —  outside  diameter  of  one  gear  =  D  + 

6.  o  —  outside  diameter  of  other  gear  =  d+-~ 

•*  n 

7.  T  =  number  of  teeth  marked   on   cutter   for   one   gear 


COS^O! 

8.  t  =  number  of  teeth  marked  on  cutter  for   other  gear 

N 
cos3/3 

9.  L  =  lead  of  spiral  for  one  gear  =  irD  cot  a. 
10.     /  =  lead  of  spiral  for  other  gear  =  nd  cot  ft. 

Example 
Given  or  assumed: 

i.   See  illustration.         2.  Pn  =  10.        3.   <f>  =  45  degrees. 

4.  C  —  4  inches. 

5.  N  =  CP«cos<£  =  4  X  10  X  0.70711  =  28.28,  say  28  teeth. 

To  find: 

1.  sin  2  a  —  0.98664,     or  a  =  40°  19'. 

2.  ft  =  90°  —  a  =  49°  41'. 

3.  D  = =  —    2  =  3.672  inches. 

Pn  COS  «         10  X  0.76248 

N  28 

4.  a  =  — = =  4.328  inches. 

Pn  cos  ft      10  X  0.64701 

5.  0  =  3.672  +  0.2  =  3.872  inches. 

6.  o  =  4.328  +  0.2  =  4.528  inchest 

N  28  ,, 

— i—  =  7 — 7-7-,  =  63.6,  say  64  teeth. 
cos3  a      (o.762)3 

N  28  .,•• 

— —  =  7— T — r-,  =  103.8,  say  104  teeth. 
cos3  ft      (o.647)3 

9.    L  =  7rZ>cota  =  3.1416  X  3.672  X  1.1787  =  13.597  inches. 
10.      /  =  -n-dcotft  =  3.1416  X  4.328  X  0.84841  =  11.536  inches. 

7.  Shafts  at  Right  Angles,  Ratio  Unequal  and  Center  Distance 
Approximate.  —  Here  the  only  two  terms  which  may  not  be 


RULES  AND  FORMULAS 


43 


decided  upon  at  the  outset  are  the  number  of  teeth  in  each  gear. 
Each,  of  course,  must  be  a  whole  number  and  correspond  with 
the  center  distance,  angle  of  spiral  and  normal  pitch.  Let 

N 


n 


and 


Pn  COS  a       Pn  COS  j8 


Then 


N      7? 
—  =  K, 

n 


Rn 


or      N  =  Rn. 


n 


=  2C. 


PnCOS/3 

Multiply  by  Pn  cos  a  cos  0.     Then 

Ifo  cos  ]8  +  ft  cos  a  =  2  CPn  cos  a  cos  0 
^  (.R  cos  |8  +  cos  a)  =  2  CPn  cos  a  cos  j8, 


and 


n  = 


2  CPn  cos  a  cos  ]8 


i?  cos  ]8  +  cos  a 

which  gives  the  number  of  teeth  in  the  pinion  in  terms  of  the 
center  distance,  angle  of  spiral  and  normal  diametral  pitch. 

When  a  spiral  angle  of  45  degrees  is  used,  the  last  formula 
becomes,  by  substituting  the  numerical  values  of  the  cosine  of 
both  angles,  which  is  0.70711: 

_  2  CPn  0.70711  X  0.70711 
R  X  0.70711  -f  0.70711 


or 


n  = 


i.4iCPr 


\ 


Formulas,  Case  7 

Sum  of  spiral  angles  of  gear  and  pinion  must  equal  90  degrees. 
Given  or  assumed: 


1.  Position  of  gear  having  right-  or  left-hand 

spiral,  depending  on  rotation  and  direc- 
tion in  which  thrust  is  to  be  received. 

2.  C0  =  approximate  center  distance. 

3.  Pn  =  normal  pitch  (pitch  of  cutter). 

4.  R  =  ratio  of  gear  to  pinion. 


44  SPIRAL  GEARING 

/~*    7~) 

5.  n  =  number  of  teeth  in  pinion  =  ^^ — — -for  45  degrees; 

K  +  i 

2  CaPn  COS  a  COS  (3    . 

and  — -  -  for  any  angle. 

R  cos  ]8  +  cos  a 

6.  ./V  =  number  of  teeth  in  gear  =  nR. 

7.  a  =  angle  of  spiral  on  gear.    . 

8.  /3  =  angle  of  spiral  on  pinion. 
To  find: 

(a)  When  spiral  angles  are  45  degrees. 

N 


i.    D  =  pitch  diameter  of  gear  = 


2.  d  =  pitch  diameter  of  pinion  =  •  -  - 

0.70711 

3.  0  =  outside  diameter  of  gear  =  D  +  -~ 


iy 

4.  o  =  outside  diameter  of  pinion  =  d  -\  -- 

P  n 

N 

5.  T  =  number  of  cutter  (gear)  =  - 

°-353 

6.  /  =  number  of  cutter  (pinion)  =  —  -  — 

0-353 

7.  L  =  lead  of  spiral  on  gear  =  irD. 

8.  /  =  lead  of  spiral  on  pinion  =  nd. 

9.  C  =  center  distance  (exact)  =  —  ^—~ 

2 

(b)  When  spiral  angles  are  other  than  45  degrees. 


1. 

3- 
5- 

u 

T 
L 

Pn  cos  a 

N 

2. 
4- 

6. 

a 

t 
I 

Pn  COS  0 

n 

COS3  a 
=  TrD  COt  a 

cos3/? 

=  TT^COt/3 

Given  or  assumed: 

i.   See  illustration.  2.  Ca  =3.2  inches. 

3.    Pn  =  10.  4.    R  =  1.5. 

1.41  CaPn      1.41  X  3.2  X  10 
5.      ».-— 5-L_.. sayxSteeth. 


RULES  AND   FORMULAS  45 

6.  N  =  nR  =  18  X  1.5  =  27  teeth. 

7.  a  =  45  degrees.  8.  0  =  45  degrees. 
To  find: 

1.  D  = ^-—  = ?1 =  3.818  inches. 

0.70711  Pn      0.70711  X  10 

j  n  18  .     , 

2.  d  = —  = =  2.545  inches. 

0.70711  Pn      0.70711  X  10 

3.  0  =  D  +  •£•  =  3.818  +  — .  =  4.018  inches. 

rn  10 

4.  o  =  d  +  -^-  =  2.545  +  —  =  2.745  inches. 

rn  10 

5.  r  =  — —  =  -^-  =  76.5,  say  76  teeth. 

0-353      0-353 

6.  t  =  -^-  =  -^-  =  51  teeth. 

0-353      0.353 

7.  L  =  irD  =  3.1416  X  3.818  =  12  inches. 

8.  /  =  ird  =  3.1416  X  2.545  =  8  inches. 

n      D  +  d      3.818  +  2.545  0    .     , 

9.  C  =  — ! —  =  fl ! ^^  =  3.182  inches. 

2  2 

8.  Shafts  at  Right  Angles,  Ratio  Unequal  and  Center  Distance 
Exact.  —  This  case  is  met  with  when  two  spiral  gears  are  to  re- 
place two  bevel  gears,  or  when  the  conditions  of  the  design  de- 
mand an  exact  center  distance  and  unequal  ratio.  The  normal 
pitch,  the  ratio  of  number  of  teeth  in  large  to  small  gear,  the 
exact  center  distance  and  the  approximate  spiral  angle  a  of  the 
large  gear  are  all  given  or  assumed.  Then  the  number  of  teeth 
in  the  small  gear  is  found  from  the  formula: 

2  CPn  sin  a. 

H  =  

R  tan  a  +  i 
which  is  found  as  follows: 

Let 

N       ,        n       _zC 

Pn  COS  a         Pn  COS  |8 

or  twice  the  center  distance,  or 


Pn  cos  a      Pn  sin  a 


46  SPIRAL  GEARING 

Let  *  =  *,    or    N=Rn. 

n 

Then  Rn  n  r 

™ ~i~   r»       •      ~~  —  2  L/. 

Pn  cos  a      Pn  sin  a 
Multiply  by  Pn  sin  a  cos  a.    Then 

ifo  sin  a  +  n  cos  a  =  2  CPn  sin  a  cos  a, 
ffc  (^  sin  a  +  COS  a)  =  2  CPn  sin  a  COS  a, 

or  2  CPn  sin  a  cos  a 

'ft  == . 

R  sin  a  +  cos  a 

Divide  by  cos  a.    Then 

2  CPr,  sin  a 


n  = 


R  tan  a  +  i 


2  CP 

The  formula  R  sec  a  +  cosec  a  = *,  which  is  used  in 

n 

finding  the  exact  spiral  angle,  is  found  in  the  same  manner  as  in 
some  of  the  preceding  cases.     Let, 

N      +  TT^ —  =  2  C,   and   -  =  R,   or   N  =  Rn. 


PnCOSa         PnCOSj8  U 

Then  Rn         _     n 

Pn  cos  a.      Pn  sin  a 

p 
Multiplying  by  --  we  have: 

n 

R  I  2CPn 


cos  a      sin  a 
or 


R  sec  a  +  cosec  a 


This  exact  spiral  angle  is  found  by  trial,  by  substituting  values 
found  in  a  table  of  secants  and  cosecants  in  the  equation  after 
the  proper  value  of  the  last  member  in  the  equation  has  been 
found  from  known  values.  About  45-degree  angles  will  prob- 
ably be  the  most  used,  unless,  for  some  reason  of  design,  the 
spiral  angle  of  one  gear  must  be  greater  than  that  of  the  other. 
In  using  trigonometric  tables  to  find  values  to  satisfy  the  equa- 
tion given,  use  first  tenths  only  for  trial,  then  hundredths,  and 


RULES  AND  FORMULAS 


47 


so  on,  as  the  required  value  is  approached.  This  shortens  the 
work  considerably.  In  solving  this  last  equation,  a  table  of 
functions,  giving  values  to  minutes,  is  necessary,  as  the  value 
of  its  right-hand  member  must  be  to  thousandths  of  an  inch. 

Formulas,  Case  8 
Gears  have  same  direction  of  spiral.    The  sum 

of  the  spiral  angles  will  equal  90  degrees. 
Given  or  assumed: 


1.  Position  of  gear  having  right-  or  left-hand 

spiral  depending  on  rotation  and  direc- 
tion in  which  thrust  is  to  be  received. 

2.  Pn  =  normal  pitch  (pitch  of  cutter). 

3.  R  =  ratio  of  number  of  teeth  in  large 
gear  to  number  of  teeth  in  small  gear. 

4.  aa  =  approximate  spiral  angle  of  large  gear. 

5.  C  =  exact  center  distance. 
To  find: 

1.  n  =  number  of  teeth  in  small  gear  nearest 

2  CPn  SJn  Pig 

i  +  R  tan  aa 

2.  N  =  number  of  teeth  in  large  gear  =  Rn. 

3.  a  —  exact  spiral  angle  of  large  gear,  found  by  trial  from 

2  CPn 


R  sec  a  -\-  cosec  a.  = 


n 


4- 
5- 

6. 


8. 
9- 


=  exact  spiral  angle  of  small  gear  =  90°  —  a. 

N 


D  =  pitch  diameter  of  large  gear  = 
d  =  pitch  diameter  of  small  gear  = 


Pn cos  a 

n 

PnCOS/3 


O  =  outside  diameter  of  large  gear  =  D  -f  — 

Pn 

o  =  outside  diameter  of  small  gear  =  d  +  — 

•Ln 

T  =  number  of   teeth  marked  on  cutter  for   large  gear 

N 
cos3  a 


48  SPIRAL  GEARING 

10.  /  =  number  of   teeth  marked   on  cutter   for  small  gear 

n 
~  cos3  0 

11.  L  =  lead  of  spiral  on  large  gear  =  irD  cot  a. 

12.  I  =  lead  of  spiral  on  small  gear  =  ird  cot/3. 

Example 

Given  or  assumed: 

1.  See  illustration          2.   Pn  =  8.  3.   R  =  3. 
4.   aa  =  45  degrees.       5.   C  =  10  inches. 

To  find: 

2  CPn  sin  «a        2  X  10  X  8  X  0.70711 

!.«.*•-  —  —       -  =   -  —  *-*  —  =  28.25,     say 

I  +R  tan  aa  i  +  3 

28  teeth. 

2.  N  =  En  =  3  X  28  =  84  teeth. 

2  CPn          2X10X8  fc0~     , 

3.  A  sec  «  +  cosec  a  =  -  =  •  --  =5.714,  or  a  =46   6. 

W  2o 

4.  j3  =  90°  -  a  =  90°  -  46°  6'  =  43°  54'. 

N  84  •     i. 

5.  D  =  —  -  =  -  -  *;  -  =  15.143  inches. 

8X0.6934 


>   :  j  n  28  0      .     , 

6.  a  =  —  -  =  -  -  =  4.857  inches. 

Pncos|3      8X0.72055 

7.  0  =  D  +  -^  =  15.143  +  o-25  =  !5-393  inches. 

*» 

2 

8.  o  =  d  +  ^~  =  4-857  +  0.25  =  5.107  inches. 

*  n 

9.  T  =  —  ~  =  —  —  =  say  252  teeth. 

cos3  a      0.333 

10.  /  =  —  :    -  =  -  =  say  75  teeth. 

cos3/?      0.374 

11.  L  —  irDcota  =  3.1416  X  15.143  X  0.96232  =  45.78  inches. 

12.  /  =  TrJcotjS  =  3.1416  X  4.857  X  1.0392  =  15.857  inches. 

Shafts  at  a  45-degree  Angle.  —  In  the  following  four  cases 
formulas  will  be  given  for  calculating  spiral  gears  with  a  shaft 
angle  of  45  degrees.  As  seen  in  Fig.  31  the  treatment,  as  far  as 
the  design  is  concerned,  will  be  the  same  for  a  i35-degree  shaft 


RULES  AND   FORMULAS 


49 


\ 


angle  as  for  an  angle  of  45  degrees.  Thrust  diagrams,  Figs.  13 
to  28,  are  given  as  an  aid  in  determining  the  direction  of  thrust 
and  rotation,  and  the  direction  of  spiral,  whether  right-  or  left- 
hand.  The  arrows  shown  indicate  the  direction  of  the  reaction 
against  the  thrust  caused  by  the  tooth  pressure. 

The  relation  between  the  direction  of  rotation,  direction  of 
spiral  and  spiral  angle  may  be  studied  in  Figs.  32  and  33.  In 
Fig.  32  two  spiral  gears  are  shown,  one  in  front  of  the  other, 
with  shafts  at  an  angle  of  45  degrees  to  each  other.  Line  AB 
represents  a  right-hand 
spiral  tooth  on  the  front 
side  of  gear  C.  Assume 
gear  C  to  rotate  in  the 
direction  shown;  then 
when  the  tooth  A  B  reaches 
the  rear  side,  it  will  be 
represented  by  line  EF, 
which  also  represents  the 
tooth  direction  on  the 
front  side  of  gear  G.  Angle 
BOH  equals  angle  EOH, 
and  it  will  be  seen  directly 
that  the  spiral  angle  of 
either  gear  equals  45  de- 
grees minus  the  spiral 
angle  of  the  other,  both 
gears  being  right-hand.  In  Fig.  33  the  spiral  angles  are  shown 
to  be  of  opposite  hand,  and  one  spiral  angle  is  45  degrees  plus 
the  other  angle.  From  these  illustrations  we  may  draw  the 
following  conclusions  relative  to  gears  with  a  shaft  angle  of  45 
degrees : 

When  the  spiral  angle  of  either  gear  is  less  than  45  degrees, 
then  the  spiral  angles  are  the  same  hand,  and  one  spiral  angle 
is  45  degrees  minus  the  other.  When  the  spiral  angle  of  either 
gear  is  greater  than  45  degrees,  then  the  spiral  angles  are  of 
opposite  hand,  and  the  spiral  angle  of  one  gear  is  45  degrees 
plus  the  spiral  angle  of  the  other. 


Machinery 


Fig.  31.  Diagrammatical  View  showing  Gen- 
eral Arrangement  of  Spiral  Gearing  with 
Shafts  at  a  45-degree  Angle 


SPIRAL  GEARING 


9.  Shafts  at  a  45-degree  Angle,  Ratio  Equal  and  Center  Dis- 
tance Approximate.  —  As  already  stated,  the  spiral  angle  of  one 
gear  must  equal  45  degrees  plus  or  minus  the  spiral  angle  of  the 
other.  The  formulas  in  Part  (a)  in  the  following  are  to  be  used 
when  the  spiral  angles  of  both  gears  are  22  J  degrees,  which  will 
often  be  the  case.  The  pitch  diameter  of  both  gears  will  be 
equal,  and  are  found  by  the  formula  (see  below  for  notation): 

N  N 


Further 


D  = 


T  = 


PnCOS22f° 

N 


0.92388  Pn 

N 


COS 


0.788 


L  =  7rZ>COt22j°  =  7.584!). 


Machinery 


Figs.  32  and  33. 


Relation  between  the  Spiral  Angles  of  Teeth  in 
Two  Gears 


The  other  formulas  are  the  same  as  those  given  in  the  preced- 
ing cases.     Part  (b)  is  used  for  unequal  spiral  angles. 

Formulas,  Case  9 

The  sum  of  the  spiral  angles  of  the  two  gears 
equals  45  degrees,  and  the  gears  are  of  the 
same  hand,  if  each  angle  is  less  than  45 
degrees.  The  difference  between  the  spiral 
angles  equals  45  degrees,  and  the  gears  are 
of  opposite  hand,  if  either  angle  is  greater  than  45  degrees. 
Given  or  assumed: 

i.  Hand  of  spiral,  depending  on  rotation  and  direction  in 
which  thrust  is  to  be  received. 


RULES  AND  FORMULAS  51 

2.  Ca  =  approximate  center  distance. 

3.  Pn  =  normal  pitch  (pitch  of  cutter). 

4.  a  =  angle  of  spiral  of  driving  gear. 

5.  /3  =  angle  of  spiral  of  driven  gear. 

s         *r  u  r   4.  4.  2  C*Pn  COS  a  COS  j8 

6.  A'  =  number  of  teeth  nearest — 

cos  a  +  cos  /3 

To  find: 

(a)  When  spiral  angles  are  22!  degrees. 

N 

1.  D  =  pitch  diameter  =  — 

0.9239  Pn 

2 

2.  0  =  outside  diameter  =  D  +  — 

3.  T  =  number  of  teeth  marked  on  cutter  =  N+  0.7880 

4.  L  =  lead  of  spiral  =  7.584!). 

5.  C  =  center  distance  =  ZX 

(&)  When  spiral  angles  are  other  than  22^  degrees. 

N 

i.    D  —  pitch  diameter  of  driver  =  — 

Pn cos  a 

N 


2.     d  =  pitch  diameter  of  driven  gear  = 


Pncos/3 


2 

3.  0  =  outside  diameter  of  driver  =  D  +  •=- 

-t  n 

rt 

4.  o  =  outside  diameter  of  driven  gear  =  d  +  — 

4  » 

5.  T1  =  number  of  teeth  marked  on  cutter  for  driver 

=  N  -T-  COS3  a. 

6.  t  =  number  of  teeth  marked  on  cutter  for  driven 

gear  =  N  -7-  cos3  0. 

7.  Z,  =  lead  of  spiral  for  driver  =  irD  cot  a. 

8.  /  =  lead  of  spiral  for  driven  gear  =  ird  cot  0. 

9.  C  =  actual  center  distance  =  sum  of  pitch  radii. 

Example 
Given  or  assumed: 

i.  See  illustration.  2.     Ca  =  4  inches. 

3.  Pn  =  10.  4  and  5.   a  =  /3  =  22^  deg. 

6.    tf  =  37. 


52  SPIRAL  GEARING 

To  find: 

N  ^7 

1.  D  =  -     — —  =  -       **— —  =  4-005  inches. 

0.9239  Pn      0.9239X10 

2  2 

2.  0  =  D  +  —  =  4.005  H =  4.205  inches. 

3.  T  =  N  H-  0.788  =  37  -s-  0.788  =  47  teeth. 

4.  L  =  7.584!)  =  7.584  X  4-005  =  30.374  inches. 

10.  Shafts  at  a  45-degree  Angle,  Ratio  Equal  and  Center 
Distance  Exact.  —  Following  the  same  method  of  reasoning  as 
in  Case  (6),  the  approximate  number  of  teeth  in  each  gear  is 
found  from  the  equation: 

N  N  r 

=  2  C 


Pn  COS  aa       Pn  COS 

from  which,  multiplying  by  Pn  cos  a0  cos  0a, 

N  cos  j3a  +  N  cos  o!a  =  2  CPn  cos  aa  cos  j8a,  or 

^.  =  2  CPn  COS  o;a  COS  ft, 

cos  j(30  +  cos  a0 

After  the  exact  number  of  teeth  to  be  used  in  each  gear  is 
found  from  the  last  equation,  the  spiral  angles  are  found  from 
the  same  equation  used  in  finding  the  approximate  number  of 
teeth. 

N  N 

Let 1 =  2  C.  where  a  and  0  are  now  the 

Pn  cos  a      Pn  cos  0 

exact  spiral  angles.    The  secant  being  the  reciprocal  of  the  cosine, 

N  N  2  CP 

— -  sec  a  +  —  sec  0  =  2  C,  or  sec  a  +  sec  0  =  — rp 

-t  n  ^n  -^V 

By  using  a  table  of  secants,  reading  to  minutes,  angles  can  be 
found  to  satisfy  this  equation,  after  very  few  trials. 

Formulas,  Case  10 

The  sum  of  the  spiral  angles  of  the  two  gears 
equals  45  degrees,  and  the  gears  are  of  the 
same  hand,  if  each  angle  is  less  than  45  de- 
grees. The  difference  between  the  spiral 
angles  equals  45  degrees,  and  the  gears  are 
of  opposite  hand,  if  either  angle  is  greater 


than  45  degrees. 


RULES  AND  FORMULAS  53 

Given  or  assumed: 

1.  Hand  of  spiral,  depending  on  rotation  and  direction  in 

which  thrust  is  to  be  received. 

2.  Pn  =  normal  pitch  (pitch  of  cutter). 

3.  C  =  center  distance. 

4.  aa  =  approximate  spiral  angle  of  one  gear. 

5.  j30  =  approximate  spiral  angle  of  the  other  gear. 

,        ,T  £     .     '    ,  ,2  CPn  COS  aa  COS^ftj 

6.  N  =  number  of  teeth  nearest 

cos  ata  4-  cos  ft, 

To  find: 

i.   a  and  |8  =  exact  spiral  angles  found  by  trial  from  sec  a  + 


N 

2.  D  =  pitch  diameter  of  one  gear  = 

Pn COS  a 

N 

3.  d  =  pitch  diameter  of  the  other  gear  =  — • 

Pn  cos  j3 

2 

4.  0  =  outside  diameter  of  one  gear  =  D  +  — - 

•*  n 

5.  o  =  outside  diameter  of  other  gear  =  d  +  — 

•* n 

6.  T  =  number   of   teeth  marked   on  cutter   for    one    gear 

=   N  -r-  COS3  0!. 

7.  /  =  number  of   teeth  marked  on   cutter  for  other   gear 


8.  L  =  lead  of  spiral  for  one  gear  =  TT/)  cot  a. 

9.  /  =  lead  of  spiral  for  other  gear  =  ird  cot  0. 


Given  or  assumed: 

i.   See  illustration.         2.   Pn  =  8.        3.   C  =  10  inches. 


4.   aa  =  15°.  5.   &  =  30°. 


6     „  _  2  CPn  cos  aa  cos  ft,  _  2  X  io  X  8  X  0.96593  X  0.86603 
cos  aa  +  cos  ftj  0-96593  +  0.86603 

=  73  teeth. 
To  find: 


,  „  t  n      2  X  io  X  8 

i.   a  and  |8  from  seco:  +  sec/3=  -  —  —  -  =  -  =  2.1918; 

^  73 

by  trial  a  and  0,  respectively,  =  i4°44'  and  30°  16'. 


54  SPIRAL  GEARING 

N  7^ 

2.    D  = =          /0  =  9435  inches. 

8X0.96712 


S-    d  = = „,  ,    =  10.565  inches. 

Pncosj8      8X0.86369 

4.  0  =  D  +  -§•  =  9435  +  f  =  9'685  inches. 

Jrn  o 

5.  o  =  d  +  -—  =  10.565  +  f  =  10.815  inches. 

jTn  5 

6.  T  =  N  -r-  cos3  a  =  73  -r-  0.904  =  8 1  teeth. 

7.  /  =  N  -r-  cos3j8  =  73  -T-  0.645  =  JI3  teeth. 

8.  L  =  irD  cot  a  =  TT  X  9435  X  3.803  =  112.72  inches. 

9.  /  =  ird  cot  |8  =  TT  X  10.565  X  1.714  =  56.889  inches. 

ii.  Shafts  at  a  45-degree  Angle,  Ratio  Unequal  and  Center 
Distance  Approximate.  —  A  formula  for  finding  the  number  of 
teeth  in  the  small  gear  is  found  from  the  equation: 

N  n 

j ____     n  ( 

"  —p  —    Z  ^/ 

by  solving  for  the  value  of  n,  the  relation  of  N  to  n  being: 

R  =  ~  ,  or  N  =  Rn. 
n 

Then  multiplying  by  Pn  cos  a  cos  /3,  we  have: 

N  cos  0  +  n  cos  a  =  2  CPn  cos  a  cos  /3 
and  substituting  Rn  for  N: 

_  2  CPn  COS  a  COS  )8 

.R  cos  |8  +  cos  a 

After  finding  the  number  of  teeth  N  from  the  relation  Rn  =  N, 
the  pitch  diameters  are  found  in  the  manner  previously  described. 
In  Part  (a)  are  given  the  formulas  to  be  used  when  both  spiral 
angles  are  22^  degrees.  The  constants  were  found  from  the 
numerical  values  of  the  functions  in  the  formulas  of  Part  (&), 
which  latter  formulas  are  used  for  unequal  spiral  angles  in  the 
two  gears. 

Formulas,  Case  n 

The  sum  of  the  spiral  angles  of  the  two  gears  equals  45 
degrees,  and  the  gears  are  of  the  same  hand,  if  each  angle  is 


RULES  AND  FORMULAS 

less  than  45  degrees.    The  difference  between 
the  spiral  angles  equals  45  degrees,  and  the 
gears  are  of  opposite  hand,  if  either  angle  is 
greater  than  45  degrees. 
Given  or  assumed: 

1.  Hand   of   spiral,  depending  on   rotation 

and  direction  in  which  thrust  is  to  be  received. 

2.  Ca  =  center  distance. 

3.  Pn  =  normal  pitch  (pitch  of  cutter). 

4.  R  =  ratio  of  gear  to  pinion,  N  -5-  n. 

5.  a  =  angle  of  spiral  on  gear. 

6.  /3  =  angle  of  spiral  on  pinion. 

-  ^     xl_  .        .    .  ,  2  CgPn  COS  a  COS  /3 

7.  n  —  number  of  teeth  in  pinion  nearest—^ —  • 

R  cos  |8  +  cos  a 

8.  N  =  number  of  teeth  in  gear  =  Rn. 

To  find: 

(a)  When  a  =  0  =  22^  degrees. 

N 

i.    D  =  pitch  diameter  of  gear  = 


55 


2.     d  =  pitch  diameter  of  pinion  = 


0.9239  Pn 
n 


0.9239 Pn 


3.  O  —  outside  diameter  of  gear  =  D  +  — 

•*  n 

4.  0  =  outside  diameter  of  pinion  =  d  +  •— 

in 

5.  T  =  number  of  teeth  marked  on  cutter  for  gear 

=  N  -f-  0.788. 

6.  /  =  number  of  teeth  marked  on  cutter  for  pinion 

=  n  -f-  0.788. 

7.  L  =  lead  of  spiral  on  gear  =  7.584!). 

8.  /  =  lead  of  spiral  on  pinion  =  7.584^. 

9.  C  =  actual  center  distance  = 

2 

(6)  When  a  and  0  are  any  angles. 

i.    Z)  =  pitch  diameter  of  gear  = 


56  SPIRAL  GEARING 

2.  d  =  pitch  diameter  of  pinion  = 

Pncos/3 

2 

3.  0  =  outside  diameter  of  gear  =D  -\ 

•*  n 

4.  o  =  outside  diameter  of  pinion  =  d  +  • — 

*n 

5.  T1  =  number  of  teeth  marked  on  cutter  for  gear 

=  N  -r-  COS3  a. 

6.  t  =  number  of  teeth  marked  on  cutter  for  pinion 

=  n  -T-  cos3  /3. 

7.  L  =  lead  of  spiral  on  gear  =  wD  cot  a. 

8.  /  =  lead  of  spiral  on  pinion  =  ird  cot  /3. 

9.  C  =  actual  center  distance  =  - 

Example 
Given  or  assumed: 

i.   See  illustration.        2.   C  =  12  inches.         3.  Pn  =  6. 

4.     j£  =  3.  5.   a  =  20  deg.  6.   j8   =  25  deg. 

_  2  CPn  cos  a  cos  j8  _  2  X  12  X  6  X  0.93969  X  0.90631 
^'  J£  cos  0  +  cos  a  (3  X  0.90631)  +  0.93969 

=  34  teeth,  approx. 
8.    N  =  Rn  =  3  X  34  =  102  teeth. 
To  find: 

AT  IO2 

1.  Z)  =  • =  —  =  18.091  inches. 

Pn  cos  a      6  X  0.93969 

2.  d  =  — —  = 34  ,      =  6.252  inches. 

Pncos/3      6X0.90631 

3.  0  =  Z>  +  ~  =  18.091  H- 1  =  18.424  inches. 

-t  n  O 

o  ^ 

4.  o  =  d  H-  —  =  6.252  +  -  =  6.585  inches. 

./ n  O 

5.  T   =   N  -r-  COS3  a  =   102   -r-  0.83   =   123  teeth. 

6.  t  =  n  -r-  cos3 18  =  34  -r-  0.744  =  46  teeth. 

7.  L  =  irD  cot  a  =  TT  X  18.091  X  2.747  =  156.12  inches. 

8.  /  =  TrdcotjS  =  TT  X  6.252  X  2.145  =  42.13  inches. 

D -\- d      18.001+6.252  .     , 

o.    C  =  =  — ^ — ! *-  =  12.1715  inches. 


RULES  AND  FORMULAS  57 

12.  Shafts  at  a  45-degree  Angle,  Ratio  Unequal  and  Center 
Distance  Exact.  —  This  case  could  be  used  under  the  same  con- 
ditions spoken  of  in  Case  (8).  The  number  of  teeth  is  found 
exactly  as  in  Case  (n),  after  which  the  exact  spiral  angles  are 
found  by  trial  from  the  equation: 


n 

which,  in  turn,  is  found  from  the  equations 

•5-^—  +  ^T—»  =  2C,  and  N  =  En 
Pn  cos  a      Pn  cos  p 

in  the  same  manner  as  before.  When  using  these  equations  for 
finding  spiral  angles,  the  trigonometrical  tables  used  must  give 
values  to  minutes  in  order  to  insure  accuracy. 

Formulas,  Case  12 

The  sum  of  the  spiral  angles  of  the  two  gears 
equals  45  degrees,  and  the  gears  are  of  the 
same  hand,  if  each  angle  is  less  than  45  de- 
grees. The  difference  between  the  spiral 
angles  equals  45  degrees,  and  the  gears  are 
of  opposite  hand,  if  either  angle  is  greater 
than  45  degrees. 
Given  or  assumed: 

1.  Hand  of  spiral,  depending  on  rotation  and  direction  in 

which  thrust  is  to  be  received. 

2.  Pn  —  normal  pitch  (pitch  of  cutter). 

3.  R  =  ratio  of  large  to  small  gear  =  N  -5-  n. 

4.  aa  =  approximate  spiral  angle  of  large  gear. 

5.  j80  =  approximate  spiral  angle  of  small  gear. 

6.  C  =  center  distance. 

7  .      n  =  number  of  teeth  in  small  gear  nearest  -  -  -  —  -  — 

R  COS  fa  +  COS  aa 

8.     N  =  number  of  teeth  in  large  gear  =  Rn. 

To  find: 

i.    a  and  |8,  exact  spiral  angles,  by  trial  from  R  sec  a  +  sec  P 

2CPn 


r\\ 


58  SPIRAL  GEARING 

N 


2.  D  =  pitch  diameter  of  large  gear  = 

3.  d  =  pitch  diameter  of  small  gear  = 


Pn cos  a 

n 

PnCOS/3 
2 


4.  O  =  outside  diameter  of  large  gear  =  D 

•L-  n 

sy 

5.  o  =  outside  diameter  of  small  gear  =  d  -+-  —  - 

•*  n 

6.  T  =  number  of  teeth  marked  on   cutter   for   large  gear 

=  N  -T-  COS3  a. 

7.  /  =  number  of  teeth  marked  on  cutter  for  small  gear 

=  n  -7-  cos3  ]8. 

8.  Z,  =  lead  of  spiral  for  large  gear  =  irD  cot  a. 

9.  /  =  lead  of  spiral  for  small  gear  =  ird  cot  /3. 

Example 
Given  or  assumed: 

i.   See  illustration.        2.  Pn  =  4.  3.  J2  =  4. 

4.  aa  =  50  degrees.        5.   0«  =  5  degrees.     6.   C  =  30  inches. 

_  2CPnCOSa;aCOS)8o  __  2X30X4X0.643X0.996  _  , 

"  ~       (4  X  0.996)  +0.643 


8.  N  =  Rn  =  4  X  33  =  132  teeth. 
To  find: 

1.  aand/3from£seca  +  sec/3  =  ^^-n  =  2X3oX4  =  7.273; 

n  33 

by  trial  a  =  50°  21',  and  j8  =  5°  21'. 

2.  D  =  —  -  =  -  ^—  —  =  51.716  inches. 

Pncoso:      4X0.63810 

3.  d  =  n  n       =  -  M—  —  =  8.286  inches. 

Pn  cos  18      4  X  0.99564 

sy  sy 

4.  0  =  D  -f  —  =  51.716  +  -  =  52.216  inches. 

-Tn  4 

5.  o  =  d  +  ~  =  8.286  +  -  =  8.786  inches. 

rn  4 

6.  T  =  N  +  cos3  a  =  132  -J-  0.26  =  508  teeth. 

7.  t  =  n  +  cos3  /3  =  33  -T-  0.987  =  33  teeth. 

8.  L  =  TrDcota  =  TT  X  51.716  X  0.82874  =  134.6  inches. 

9.  I  =  7rdcot/3  =  TT  X  8.286  X  10.678  =  278  inches. 


RULES  AND   FORMULAS 


59 


Spiral  Gears  with  Shafts  at  Any  Angle.  —  When  designing 
spiral  gears  with  shafts  at  an  angle  other  than  90  degrees  to 
each  other,  it  is  of  considerable  advantage  to  draw  the  outline 
of  one  gear  on  a  piece  of  drawing  paper  tacked  to  the  board,  and 
the  outline  of  the  other  on  a  piece  of  tracing  paper,  as  indicated 
in  the  accompanying  engraving,  Fig.  34.  In  this  way  the  gear 
drawn  on  the  tracing  paper  can  be  moved  about  to  thp  correct 
angle  with  relation  to  the  gear  beneath,  and  the  conditions  of 


\ 


Machinery 


Fig.  34.     Method  of  using  Tracing  Paper  for  Spiral  Gear  Problems 

thrust,  direction  of  rotation  and  hand  of  spiral  can  be  more 
easily  determined.  The  thrust  diagrams,  Figs.  13  to  28,  apply 
also  to  the  gears  at  present  dealt  with.  With  the  shafts  at  any 
given  angle,  the  sum  of  the  spiral  angles  of  the  two  gears  must 
equal  the  angle  between  the  shafts,  and  the  spiral  must  be  of 
the  same  hand  in  both  gears,  if  each  spiral  angle  is  less  than  the 
shaft  angle;  but  if  the  spiral  angle  of  one  of  the  gears  is  greater 
than  the  shaft  angle,  then  the  difference  between  the  spiral  angles 
of  the  two  gears  will  be  equal  to  the  shaft  angle,  and  the  gears 
will  be  of  opposite  hand. 


60  SPIRAL  GEARING 

Detailed  explanation  of  the  derivation  of  the  formulas  in  the 
following  four  cases  is  unnecessary,  as  these  are  arrived  at  in  a 
manner  similar  to  that  referred  to  in  the  previous  cases.  It 
may  be  mentioned,  however,  with  relation  to  Case  (15)  that  the 
formula  for  the  number  of  teeth  in  the  smaller  gear,  in  the  case 
when  both  have  the  same  spiral  angles,  is  found  by  substituting 
cos  a  for  cos  j8  in  the  formula: 

2  CgPn  cos  a  cos  |8 
R  cos  |8  +  cos  a 

from  which  we  get  the  number  of  teeth  in  the  pinion: 

2  CaPn  COS  a  COS  a  _  CgPn  COS  a 

R  cos  a  +  cos  a  R  +  i 

In  Case  (16)  it  sometimes  happens,  after  the  exact  spiral 
angles  a  and  /3,  and  the  corresponding  pitch  diameters  have  been 
determined,  that  the  center  distance  does  not  come  exactly  as 
required,  within  a  few  thousandths  inch.  Theoretically  it  would 
then  be  necessary  to  alter  the  spiral  angles  found  from  one- 
quarter  to  one-half  a  minute,  in  order  that  the  center  distance 
may  figure  out  correctly.  However,  this  refinement  is  of  doubt- 
ful practical  value,  as  it  would  be  impossible  to  set  the  machine 
on  which  the  gears  are  to  be  cut  to  such  a  minute  sub-division 
of  a  degree. 

13.  Shafts  at  any  Angle,  Ratio  Equal,  Center  Distance  Approx- 
imate. —  The  sum  of  the  spiral  angles  of  the  two  gears  equals 
the  shaft  angle,  and  the  gears  are  of  the  same 
hand,  if  each  angle  is  less  than  the  shaft 
angle.  The  difference  between  the  spiral 
angles  equals  the  shaft  angle,  and  the  gears 
are  of  opposite  hand,  if  either  angle  is  greater 
than  the  shaft  angle. 
Given  or  assumed: 

1.  Hand  of  spiral,  depending  on  rotation  and  direction  in 

which  thrust  is  to  be  received. 

2.  Ca  =  approximate  center  distance. 

3.  Pn  =  normal  pitch  (pitch  of  cutter). 

4.  a  =  angle  of  spiral  of  one  gear. 


RULES  AND  FORMULAS  6l 

5.  /3  =  angle  of  spiral  of  other  gear. 

*  mr  r  4.  *   2  Ca^n  COS  «  COS  |8 

6.  TV  =  number  of  teeth  nearest  -  — - 

cos  a  +  cos  /3 

To  find: 

1.  Z)  =  pitch  diameter  of  one  gear  =  — 

Pn  cos  a 

N 

2.  d  =  pitch  diameter  of  other  gear  =  — — 

3.  O  =  outside  diameter  of  one  gear  =  D  +  — - 

P n 

4.  o  =  outside  diameter  of  other  gear  =  d  +  -—• 

5.  T  =  number  of   teeth  marked   on   cutter  for   one   gear 

=  TV  -7-  COS3  a. 

6.  t  =  number  of  teeth  marked  on  cutter   for  other  gear 

7.  L  =  lead  of  spiral  for  one  gear  =  irD  cot  a. 

8.  /  =  lead  of  spiral  for  other  gear  =  ird  cot  /3. 

9.  C  =  actual  center  distance  =  — 


Example 

Given  or  assumed  (angle  of  shafts,  30  degrees) : 

i.   See  illustration.         2.   Ca  =  5  inches.       3.   Pn  =  10. 
4.   a  =  20  degrees.         5.     /3  =  10  degrees.  6.   N  =  48. 

To  find: 

r,  N  48  0  .      , 

1.  D  =  — = :n =  5.108  inches. 

Pn  cos  a      10  X  0.9397 

TV  48 

2.  d  =  -—        -  = ^ — — -  =  4.874  inches. 

Pn  cos  j8       10  X  0.9848 

3.  O  =  D  +  -J-  =  5.108  +  —  =  5.308  inches. 

-T n  IO 

4.  o  =  d  +  —  =  4.874  H =  5.074  inches. 

5.  T  =  N  -^  cos3  a  =  48  -T-  0.83  =  58  teeth. 

6.  /  =  N  -T-  cos3  0  =  48  -T-  0.96  =  50  teeth. 

7.  L  =  wDcota  =  TT  X  5.108  X  2.747  =  44.08  inches. 


62  SPIRAL  GEARING 


- 
' 


8.     /  =  7r^cot/3  =  TT  X  4-874  X  5.671  =  86.84  inches. 

~      D  +  d     5.108  +  4.874 
9-    C  = =  -      ~      ~  =  4-99i  inches. 

14.  Shafts  at  Any  Angle,  Ratio  Equal,  Center  Distance 
Exact. — The  sum  of  the  spiral  angles  of  the  two 
gears  equals  the  shaft  angle,  and  the  gears  are 
of  the  same  hand,  if  each  angle  is  less  than  the 
shaft  angle.  The  difference  between  the  spiral 
angles  equals  the  shaft  angle,  and  the  gears  are 

of  opposite  hand,  if  either  angle  is  greater  than  the  shaft  angle. 

Given  or  assumed: 

1.  Hand  of  spiral,  depending  on  rotation  and  direction  in 

which  thrust  is  to  be  received. 

2.  C  =  center  distance. 

3.  Pn  —  normal  pitch  (pitch  of  cutter). 

4.  oLa  =  approximate  spiral  angle  of  one  gear. 

5.  j8a  =  approximate  spiral  angle  of  other  gear. 

6.  -N  =  number  of  teeth  nearest 2  CPn  cos  a" cos  ^a 

cos  aa  +  cos  /3a 

To  find: 

i.   a  and  13  =  exact  spiral  angles,  found  by  trial  from  sec  a  + 


N 

2.  D  =  pitch  diameter  of  one  gear  =  — 

Pn  COS  a 

N 

3.  d  =  pitch  diameter  of  other  gear  =  -  • 

Pncos/3 

r\ 

4.  0  =  outside  diameter  of  one  gear  =  D  -\  -- 

*  • 

5.  o  =  outside  diameter  of  other  gear  =  d  +  — 

*» 

6.  T  =  number  of   teeth   marked   on  cutter   for  one  gear 

=  N  -T-  COS3  a. 

7.  t  —  number  of   teeth  marked  on   cutter  for  other  gear 

=  N  -5-  cos3  18. 

8.  L  =  lead  of  spiral  for  one  gear  =  irD  cot  a. 

9.  I  =  lead  of  spiral  for  other  gear  =  ird  cot  0. 


RULES  AND   FORMULAS  63 

Example 

Given  or  assumed  (angle  of  shafts,  50  degrees)  : 

i.   See  illustration.         2.   C  =  10  inches.         3.   Pn  =  10. 
4.   aa  =  20  deg.  5.   (3a  =  30  deg. 

2  CPn  cos  «a  cos  /3a       2  X  10  X  10  X  0.93969  X  0.86603 

Q  ^y      =        •  -      =      ' 

COS  Cia  -}-  COS  /?o  0.93969  +  0.86603 

=  90  teeth. 
To  find: 

n  ,  2  CPn          2   X   10  X   10 

1.  a  and  |8  from  sec  a  +  sec  /3  =  —  —  =  -  =  2.222; 

N  90 

by  trial  a  and  0,  respectively,  =  19°  20'  and  30°  40'. 

2.  D  =  =-^  —  =  —  —  ^  —  —  =  9-537  inches. 

Pncosa       10X0.94361 

N  go  ,    .    , 

7.     d  =  -  =  -  *  -  •  =  10.463  inches. 
Pncos/3      10X0.86015 

4.  0  =  D+j-  =  9.537  +  ^  =  9-737  inches. 

5.  o  =  d  +~  =  10.463  +  —  =  10.663  inches. 

6.  T  =  N  -i-  cos3  o:  =  90  -s-  0.84  =  107  teeth. 

7.  t  =  N  -r-  cos3  18  =  90  -T-  0.64  =  141  teeth. 

-8.    L  =  irDcota  =  TT  X  9-537  X  2.85  =  85.39  inches. 
9.      /  =  TrJcotjS  =  TT  X  10.463  X  1.686  =  55.42  inches. 

15.  Shafts  at  Any  Angle,  Ratio  Unequal,  Center  Distance 
Approximate.  —  The  sum  of  the  spiral  angles  of  the  two  gears 
equals  the  shaft  angle,  and  the  gears  are  of 
the  same  hand,  if  (each  angle  is  less  than  the 
shaft  angle.  The  difference  between  the 
spiral  angles  equals  the  shaft  angle,  and  the 
gears  are  of  opposite  hand,  if  either  angle  is 
greater  than  the  shaft  angle. 

Given  or  assumed: 

1.  Hand  of  spiral,  depending  on  rotation  and  direction  in 

which  thrust  is  to  be  received. 

2.  C0  =  center  distance. 

3.  Pn  =  normal  pitch  (pitch  of  cutter). 


* 
T 


64  SPIRAL  GEARING 

4.  R  =  ratio  of  gear  to  pinion  =  N  -*•  n. 

5.  a  =  angle  of  spiral  on  gear. 

6.  /3  =  angle  of  spiral  on  pinion. 

7.  n  =  number  of  teeth  in  pinion  nearest  2  C*P»COSQ!COSft 

R  cos  |8  +  cos  a 

for  any  angle,  and       °  "°    -  when  both  angles  are  equal. 

8.  N  =  number  of  teeth  in  gear  =  Rn. 

To  find: 

N 

1.  D  =  pitch  diameter  of  gear  =  •- 

Pn  cos  a 

2.  d  =  pitch  diameter  of  pinion  =  — - — 

Pncos/3 

3.  O  =  outside  diameter  of  gear  =  Z>  +  — 

•*» 

4.  o  =  outside  diameter  of  pinion  =  d  H 

5-     T  =  number  of  teeth  marked  on  cutter  for  gear  =  N  -=-  cos3  a. 

6.  /  =  number  of  teethmarked  on  cutter  for  pinion  =  H-T-  cos3  /3. 

7.  L  =  lead  of  spiral  on  gear  =  irD  cot  a. 

8.  /  =  lead  of  spiral  on  pinion  =  ird  cot  0. 

9.  C  =  actual  center  distance  =  — — 

2 

Example 

Given  or  assumed  (angle  of  shafts,  60  degrees) : 
i.   See  illustration.         2.   Ca=  12  inches.     3.  Pn=  8. 
4-     R  =  4-                     5.    a  =  30  degrees.    6.    /3  =  30  degs. 
2  CaPn  cos  a      2  X  12  X  8  X  0.86603 
— R  4- 1  T 33  teeth. 

8.     tf  =  4  X  33  =  132  teeth. 
To  find: 


0  =  D  +  — -  =  19.052  +  -  =  19.302  inches. 

f- n  O 


RULES  AND  FORMULAS  65 

4.  o  =  d  +  T~  =  4.763  +  1  =  5-OI3  inches. 

-r»  o 

5.  T  =  N  -T-  cos3  a  =  132  -5-  0.65  =  203  teeth. 

6.  /  =  a  -i-  cos3/3  =  33  -T-  0.65  =  51  teeth. 

7.  Z,  =  irDcota  =  TT  X  19.052  X  1.732  =  103.66  inches. 

8.  I  =  TrdcotjS  =  TT  X  4.763  X  1.732  =  25.92  inches. 


~ 
C  = 


D 


d      io.cX2  +  4.763  .    , 

-  =  -*—  —  =  11-9075  inches. 


16.   Shafts  at  Any  Angle,  Ratio  Unequal,  Center  Distance 
Exact.  —  The  sum  of  the  spiral  angles  of  the  two  gears  equals 

the  shaft  angle,  and  the  gears  are  of  the  same 

hand,  if  each  angle  is   less  than   the  shaft 

angle.    The    difference    between    the    spiral 

angles  equals  the  shaft  angle,  and  the  gears 

are  of  opposite  hand,  if  either  angle  is  greater 

than  the  shaft  angle. 
Given  or  assumed: 

1.  Hand  of  spiral,  depending  on  rotation  and  direction  in 

which  thrust  is  to  be  received. 

2.  C  =  center  distance. 

3.  Pn  =  normal  pitch  (pitch  of  cutter). 

4.  aa  =  approximate  spiral  angle  of  gear. 

5.  ft,  =  approximate  spiral  angle  of  pinion. 

6.  R  =  ratio  of  gear  to  pinion  =  N  -=-  n. 

£  .       , ,     .         .    .  ,2  CPn  COS  CLa  COS  0a 

7.  n  =  number  of  teeth  in  pinion  nearest  — — — 

R  COS  /30  +  COS  Ota 

8.  N  =  number  of  teeth  in  gear  =  Rn. 

To  find: 

i.   a  and  |8,  exact  spiral  angles,  found  by  trial  from  R  sec  a 


n 

N 


N 

2.  D  =  pitch  diameter  of  gear  =  — 

Pn  COS  OL 

3.  d  =  pitch  diameter  of  pinion  = 

4.  0  =  outside  diameter  of  gear  =D  +  — 

* n 


66  SPIRAL  GEARING 

5.  o  —  outside  diameter  of  pinion  =  d  +  •— - 

•*  n 

6.  T  =  number  of  teeth  marked  on  cutter  for  gear  =  N  -f-  cos3  a. 

7.  /  =  number  of  teethmarked  on  cu  tter  for  pinion  =  w-f- cos3  0. 

8.  L  =  lead  of  spiral  on  gear  =  irD  cot  a. 

9.  /  =  lead  of  spiral  on  pinion  =  ird  cot  ft. 

Example 
Given  or  assumed  (angle  of  shafts,  60  degrees) : 

i.   See  illustration.         2.    C  =  40  inches.        3.  Pn  =  4. 
4.   aa  =  20  degrees.       5.   fta  =  40  degrees.       6.    J?  =  3. 
_  2  CPn  cos  <*a  cos  ffa  =  2  X  40  X  4  X  0.9397  X  0.766 
R  cos  fta  +  cos  aa  (3  X  0.766)  +  0.9397 

=  71  teeth. 
8.    #  =  Rn  =  3  X  71  =  213  teeth. 

To  find: 

1.  a  and  0from #seca+sec/3  =  ^^  =  2  X 4°  X 4  =  4.507; 

n  71 

by  trial  a  =  22°  24'  30"  and  ft  =  37°  35'  30". 

N  213 

2.  D  —  -=—  —  = " — —  =  57.599  inches. 

Pn  cos  a      4  X  0.92449 

j  n  71  .     , 

V     a  =  — = ~  =  22.401  inches. 

Pncos(3      4X0.79238 

4.  O  =  Z>  +  —  =  57.599  +  -  =  58-099  inches. 

*n  4 

,     .       2  .2  .       , 

5.  o  =  d  +  —  =  22.401  +  -  =  22.901  inches. 

Pn  4 

6.  T  =  N  -s-  cos3  a  =  213  -s-  0.79  =  270  teeth. 

7.  /  =  w  -r-  cos3  ft  =  71  -T-  0.497  =  J43  teeth. 

8.  L  =  TT!)  cot  a  =  TT  X  57.599  X  2.4252  =  438.8  inches. 

9.  /  =  TrJcotjS  =  TT  X  22.401  X  1.2989  =  91.41  inches. 

Special  Case  of  Spiral  Gear  Design.  —  The  following  method 
is  used  when  the  distance  between  the  centers  of  the  shafts,  the 
speed  ratio  and  an  approximate  ratio  of  the  pitch  diameters  of 
the  gears  are  given.  (Shafts  at  90  degrees  angle.)  In  the 
formulas,  let: 


RULES  AND  FORMULAS  67 

D  =  diameter  of  driver; 

d  =  diameter  of  driven  gear; 

5  =  speed  of  driver; 

5  =  speed  of  driven  gear; 
Pn  =  normal  diametral  pitch; 

a  =  angle  of  teeth  in  driver  with  its  axis; 
N  =  number  of  teeth  in  driver; 

n  =  number  of,  teeth  in  driven  gear;~ 

C  =  center  distance. 

Assume  trial  values  for  D  and  d\  then  an  approximate  angle  a 
is  derived  from  the  formula: 

ds  f  N 

—  -cot  a.  (l) 

Then  find  by  trial  the  number  of  teeth  for  each  of  the  gears 
which,  with  the  given  speed  ratio,  will  most  nearly  satisfy  the 
equation  : 

C  —       ^       4-       n  (  ") 

Pn  cos  a      Pn  sin  a 

Then  make  corrections  of  the  angle  a  until  a  value  is  found 
which  exactly  satisfies  the  last  equation.  This  being  done,  the 
pitch  diameters  are: 


d  =  —  -  —  (4) 

Pnsina 

Example.  —  Find  the  diameters  and  angles  of  teeth  of  two 
spiral  gears  with  shafts  at  right  angles;  the  distance  between 
the  centers  is  4^  inches,  the  speed  ratio  of  the  driver  to  the  fol- 
lower is  2  to  i,  and  the  ratio  of  D  to  d  is  about  9  to  8. 

Following  the  method  outlined: 

ds       8  X  i  ,  ,,0 

—  =  —  —  =  0.444  =  cot  66  ,  approx. 

By  trial  it  will  be  found  that  14  and  28  teeth  will  nearly  sat- 
isfy Equation  (2).  Substituting  these  numbers  of  teeth  and 
the  functions  of  66  degrees  in  this  equation,  we  have: 


68  SPIRAL  GEARING 

14  ,  28 

8  X  0.4067      8  X  0.9135  ~ 

Subtracting  8.  2  50  —  8.134  =  0.116. 

We  see  that  the  angle  of  66  degrees  introduces  an  error  of  0.116 
inch  for  twice  the  distance  between  centers  of  shafts.  This 
shows  that  66  degrees  is  not  exactly  the  required  angle.  Trying 
66  degrees  50  minutes,  we  have: 

14  __  __  28  __  ~ 
8  X  0.3934      8  X  0.9194 

Subtracting  8.  2  54  —  8.250  =  0.004. 

We  see  that  66  degrees  50  minutes  is  very  close  to  the  required 
angle,  as  the  error  for  twice  the  distance  between  the  centers  of 
the  shafts  is  now  only  0.004  inch;  trying  an  angle  of  66  degrees 
48  minutes,  we  have: 

-  H  -  +  —  *§  _  -  =  82.0 
8  X  0.3939      8  X  0.9191 

The  angle  of  66  degrees  48  minutes  gives  exactly  the  required 
distance  between  centers.  We  can  now  use  this  angle  in  deter- 
mining the  required  diameter  for  the  driver  and  follower  by  sub- 
stitution in  Equations  (3)  and  (4). 


=  4442 

=  3'8°8  inches" 


By  reference  to  a  table  of  natural  functions,  we  find  that  sine 
66  degrees  48  minutes  equals  cosine  23  degrees  12  minutes,  and 
this  determines  the  angle  of  the  teeth  in  the  follower  as  23  de- 
grees 12  minutes. 


CHAPTER  III 
HERRINGBONE  GEARS 

Definitions  and  Types  of  Herringbone  Gears.  — -  One  of  the 
objectionable  features  of  spiral  gears  is  the  end  thrust  necessarily 
produced  when  these  gears  are  in  action.  When  spiral  gears 
transmit  motion  between  two  parallel  shafts,  this  end  thrust 
may  be  avoided  by  placing  two  spiral  gears  side  by  side,  having 
teeth  cut  in  opposite  directions,  as  indicated  at  A,  in  Fig.  i. 
This  type  of  gearing  has  been  termed  " herringbone"  gearing. 
The  placing  of  two  spiral  gears  side  by  side,  keyed  to  the  same 
shaft,  has,  however,  certain  disadvantages,  because  it  is  prac- 
tically impossible,  with  ordinary  means,  to  so  cut  the  two  gears 
that  the  two  halves  will  be  in  perfect  mesh  at  all  times,  so  that 
each  takes  one-half  the  load.  From  a  practical  standpoint 
herringbone  gears  have,  therefore,  been  less  satisfactory  than 
straight-cut  spur  gears,  because  until  recently  no  method  was 
devised  for  producing  them  with  commercial  accuracy  at  a  reas- 
onable rate  of  speed. 

In  order  to  avoid  using  two  separate  gears,  the  two  sets  of 
spiral  teeth  have  been  cut  on  the  same  blank,  a  groove  being 
formed  in  the  center  as  indicated  at  B,  Fig.  i,  and  teeth  cut  on 
each  side.  One  method  has  also  been  developed  for  cutting  the 
two  opposite  spirals  at  the  same  time,  as  indicated  at  C.  Cast 
gears,  of  course,  can  be  made  in  one  piece,  as  indicated  at  D. 
Within  the  last  decade  a  method  has  been  developed  to  a  high 
degree  of  perfection,  by  means  of  which  herringbone  gears  can 
be  cut  both  rapidly  and  with  commercial  accuracy.  The  prin- 
ciple of  this  method  is  indicated  at  E.  Herringbone  gears  made 
by  this  method  are  called  Wuest  gears,  after  the  inventor.  The 
difference  between  these  gears  and  those  of  the  ordinary  herring- 
bone type  is  that  the  teeth  of  the  former,  instead  of  joining  at 
a  common  apex  at  the  center  of  the  face  are  stepped  or  staggered 

69 


SPIRAL  GEARING 


half  of  the  pitch  apart,  and  thus  do  not  meet  at  all.  This  arrange- 
ment of  the  teeth  does  not  affect  the  action  of  the  gears,  but 
facilitates  their  commercial  production. 

One  type  of  gear,  shown  at  F}  is  known  as  the  Citroen  gear. 
In  this  type  of  gear  the  end  thrust  has  been  avoided  by  making 
the  teeth  of  a  "wavy"  form  —  a  multiple  herringbone  type. 


Machinery 


Fig.  i.     Diagrammatical  Views  of  Different  Types  of  Herringbone  Gears 

The  difficulties  of  producing  teeth  of  this  kind  are  obvious  and 
the  ordinary  herringbone  gear  is,  without  question,  the  type 
which  meets  the  requirements  better  than  any  other  method 
for  accomplishing  the  same  result. 

Cost  of  Herringbone  Gears.  —  The  herringbone  gear  as  com- 
monly made  would  cost  about  twice  as  much  as  an  equivalent 
spur  gear  and  is  somewhat  difficult  to  cut  so  that  each  gear  will 
carry  half  the  load.  These  drawbacks  have  overshadowed  its 


HERRINGBONE  GEARS  71 

advantages  for  ordinary  uses,  and  as  a  result  its  use  is  almost 
unknown  in  the  general  run  of  machinery.  The  advent  of  the 
hobbing  process  of  cutting  helical  gears  of  the  Wuest  design, 
which  can  be  cut  very  cheaply,  has  made  them  available  in  prac- 
tically every  case,  as  far  as  cost  is  concerned,  where  spur  gears 
are  used.  The  advantages  of  herringbone  gears  are  such  that 
machine  designers  cannot  afford  to  neglect  them.  The  in- 
formation given  in  the  following  is  mainly  from  a  paper  presented 
before  the  American  Society  of  Mechanical  Engineers  by  Mr. 
Percy  C.  Day. 

Requirements  of  Power  Transmitting  Mediums.  —  The  utili- 
zation of  power  constantly  calls  for  means  to  transmit  rotary 
motion  from  one  axis  to  another.  While  there  are  many  ways 
in  which  such  transmissions  may  be  produced,  the  merits  of  all 
of  them  must  be  judged  from  the  following  standards:  (a)  relia- 
bility and  freedom  from  wear  and  tear;  (6)  economy  of  outlay; 
(c)  mechanical  efficiency;  (d)  compactness;  (e)  evenness  of 
transmission,  absence  of  shock,  jar  or  vibration;  (/)  absence  of 
noise. 

Action  of  Spur  Gearing.  —  The  aim  of  all  designers  of  gearing 
is  to  transmit  rotary  motion  from  one  axis  to  another  in  a  per- 
fectly even  manner  without  variation  of  angular  velocity.  Let 
us  consider  the  action  of  a  straight  spur  pinion  driving  a  gear. 
There  are  three  distinct  phases  of  engagement: 

First  phase:  The  root  of  the  pinion  tooth  engages  the  point  of 
the  gear  tooth. 

Second  phase:  The  teeth  are  engaged  near  the  pitch  line. 

Third  phase:  The  point  of  the  pinion  tooth  engages  the  root 
of  the  gear  tooth. 

Let  us  assume  that  the  teeth  are  accurately  cut  to  involute 
form,  so  that  if  the  pinion  moves  with  even  angular  velocity  it 
will  produce  corresponding  evenness  of  motion  in  the  gear;  and 
also  that  the  pinion  has  sufficient  teeth  to  allow  the  engagement 
of  successive  teeth  to  overlap.  At  the  beginning  of  the  first 
phase,  while  the  load  is  carried  near  the  point  of  the  gear  tooth, 
that  tooth  is  subjected  to  a  maximum  bending  stress  along  its 
whole  length.  The  portion  of  the  pinion  tooth  near  the  root  is 


72  SPIRAL  GEARING 

sliding  over  the  outer  portion  of  the  gear  tooth;  that  is  to  say, 
two  metallic  surfaces  of  small  area  are  sliding  under  heavy  com- 
pression. 

The  action  during  the  second  phase  more  nearly  approaches 
ideal  conditions.  The  teeth  are  engaged  near  their  respective 
pitch  lines  and  very  little  sliding  takes  place.  During  the  third 
and  final  phase  the  pinion  tooth  is  subjected  to  a  maximum 
bending  stress,  while  the  tooth  surfaces  again  slide  over  each 
other,  this  time  with  the  outer  portion  of  the  pinion  tooth  en- 
gaging the  gear  tooth  near  its  root.  The  point  to  be  noted  is 
that  while  those  portions  of  the  mating  teeth  which  are  near  the 
pitch  lines  transmit  the  load  with  rolling  contact,  those  which 
are  more  remote  have  to  transmit  the  same  load  with  sliding 
contact.  The  inevitable  result  is  that  the  points  and  roots  of 
all  the  teeth  tend  to  wear  away  more  rapidly  than  the  portions 
near  the  pitch  lines. 

It  may  be  suggested  that  the  sliding  action  can  be  eliminated 
by  shortening  the  teeth  so  that  they  engage  only  during  the 
phase  of  rolling  contact.  This  has  been  tried  with  a  certain 
measure  of  success  in  the  stub-toothed  gear,  but  it  cannot  be 
carried  far  enough  without  curtailing  the  arc  of  contact  so  that 
continuity  of  engagement  is  lost. 

Distortions  of  gear  teeth  of  involute  form,  whether  due  to 
inaccurate  cutting  or  subsequent  wear,  give  rise  to  all  kinds  of 
trouble.  The  average  angular  velocity  may  be  uniform,  and 
yet  the  passage  of  each  pinion  tooth  through  its  brief  engage- 
ment with  the  mating  gear  may  be  accompanied  by  successive 
retardation  and  acceleration  which,  though  small  in  itself,  takes 
place  in  such  a  short  interval  of  time  that  it  may  cause  stresses 
many  tunes  greater  than  the  average  working  load  on  the  teeth. 
These  internal  stresses  are  very  difficult  to  deal  with,  because 
they  are  indeterminate.  They  cause  noise,  vibration,  crystalli- 
zation and  fracture. 

Action  of  Herringbone  Gears.  —  Herringbone  gears  completely 
overcome  all  these  difficulties,  but  only  when  they  are  accurately 
cut.  If  we  take  two  exactly  similar  pinions  with  straight  teeth 
and  place  them  side  by  side  on  one  shaft,  with  the  teeth  of  one 


HERRINGBONE  GEARS  73 

pinion  set  opposite  the  spaces  of  the  other,  then  we  have  what 
is  known  as  a  stepped-tooth  pinion.  If  this  pinion  is  meshed 
with  a  composite  gear  made  up  in  a  similar  manner,  the  action 
is  modified  so  that  there  are  always  two  phases  of  engagement 
taking  place  simultaneously.  Such  gears  are  commonly  used  for 
rolling  mill  work,  because  they  stand  up  to  heavy  shocks  better 
than  the  plain  type.  Still  better  action  can  be  secured  by  as- 
sembling a  number  of  narrow  pinions  with  the  last  of  the  series 
one  pitch  in  advance  of  the  first  and  the  others  advanced  by  equal 
angular  increments.  As  a  practical  proposition,  however,  gears 
made  on  these  lines  would  be  costly  and  difficult  to  produce. 

The  helical  gear  is  the  logical  outcome  of  the  stepped  gear 
carried  to  its  limit,  and  built  up  from  infinitely  thin  lamina- 
tions. Since  the  steps  have  merged  into  a  helix,  there  must 
be  a  normal  component  of  the  tangential  pressure  on  the  teeth, 
producing  end  thrust  on  the  shafts.  To  obviate  end  thrust  the 
helical  teeth  are  made  right-hand  on  one  side  and  left-hand  on 
the  other.  Such  gears,  as  already  stated,  are  known  as  herring- 
bone gears. 

The  fundamental  principle  of  the  action  of  herringbone  teeth 
lies  in  the  circumstance  that  all  phases  of  engagement  take  place 
simultaneously.  This  holds  good  for  every  position  of  pinion 
and  gear,  provided  only  that  the  relationship  between  pitch, 
face  width  and  spiral  angle  is  such  as  will  insure  a  complete  over- 
lap of  engagement.  Since  all  phases  of  engagement  occur  to- 
gether, it  follows  that  the  load  is  partly  carried  by  tooth  surfaces 
in  sliding  contact  and  partly  by  surfaces  in  rolling  contact.  The 
result  is  curious  and  interesting. 

Those  portions  of  the  teeth  farthest  from  the  pitch  line,  which 
engage  with  sliding  action,  tend  to  wear  away  more  rapidly 
than  the  portions  nearest  the  pitch  line;  but  the  pitch  line  por- 
tion is  always  carrying  part  of  the  load,  and  the  effect  of  wear 
on  the  ends  of  the  teeth  merely  tends  to  throw  more  load  on 
the  center  portions;  in  other  words  there  is  a  tendency  to  con- 
centrate the  load  near  the  pitch  lines.  The  ends  of  the  teeth, 
instead  of  wearing  away  to  an  ever-increasing  extent  from  their 
original  involute  form,  are  relieved  of  some  of  the  load  from  the 


74  SPIRAL  GEARING 

moment  that  wear  commences  to  take  place.  As  soon  as  the 
load  on  these  ends  has  been  partially  relieved  and  transferred 
to  the  middle  portion,  the  wear  becomes  equalized  all  over  the 
teeth  and  they  do  not  tend  to  distort  further  from  their  original 
shape. 

It  is  quite  clear  that  an  unmeasurable  amount  of  wear  on  the 
tooth  ends  will  be  sufficient  to  relieve  them  of  all  the  load,  so 
that  the  distortion  from  the  original  form  will  be  practically 
nothing.  The  minute  extra  wear  that  does  take  place  at  the 
ends  is  only  the  amount  necessary  to  transfer  a  certain  propor- 
tion of  the  load  near  the  pitch  lines,  so  that  the  wear  is  equalized 
all  over  the  surface  of  the  teeth,  those  portions  in  sliding  con- 
tact carrying  less  than  those  in  rolling  contact. 

Advantages  Gained  by  Herringbone  Gears.  —  As  the  teeth 
keep  their  involute  form,  motion  is  transmitted  from  the  pinion 
to  the  gear  in  an  even  manner,  without  jar,  shock  or  vibration. 
Although  herringbone  teeth  may  not  be  intrinsically  stronger 
than  straight  teeth,  the  elimination  of  shock  renders  them  capable 
of  transmitting  heavier  loads.  Since  all  phases  of  engagement 
occur  simultaneously,  the  transference  of  the  load  from  one 
pinion  tooth  to  the  next  takes  place  gradually  instead  of  sud- 
denly. This  is  the  second  principle  of  herringbone  gearing,  and 
may  be  termed  continuity  of  action. 

In  straight  gears  the  continuity  of  action  is  a  function  of  the 
number  of  teeth  in  the  pinion.  In  herringbone  gears  continuity 
depends  on  the  relationship  between  the  face  width  and  the 
number  of  teeth  in  the  pinion.  Pinions  with  as  few  as  five  teeth 
have  been  used  with  success  by  merely  increasing  the  face  width 
to  suit  such  extreme  conditions.  This  feature,  which  is  peculiar 
to  herringbone  gears,  has  made  practical  the  adoption  of  ex- 
tremely high  ratios  of  reduction  hitherto  considered  impossible. 

The  third  principle  of  herringbone  gearing  is  that  the  bend- 
ing stress  on  the  teeth  does  not  fluctuate  from  maximum  to 
minimum  as  in  straight  gears,  but  remains  always  near  the  mean 
value.  This  feature  is  of  special  importance  in  rolling-mill 
driving  and  work  of  a  similar  nature. 

To  summarize  the  foregoing  .statements:   The  action  of  her- 


HERRINGBONE  GEARS  75 

ringbone  gears  is  continuous  and  smooth;  there  is  no  shock  of 
transference  from  tooth  to  tooth;  the  teeth  do  not  wear  out  of 
shape;  the  bending  action  of  the  load  on  the  teeth  is  less  than 
with  straight  gearing  and  does  not  fluctuate  to  anything  like 
the  same  extent;  the  gears  work  silently  and  without  vibration; 
back-lash  is  absent;  friction  and  mechanical  losses  are  reduced 
to  a  minimum;  herringbone  gears  can  be  used  for  higher  ratios 
and  greater  velocities  than  any  other  kind. 

Production  of  Herringbone  Gears.  —  Herringbone  gears  may 
be  produced  in  a  variety  of  ways  which  differ  from  each  other 
as  widely  as  the  character  of  the  product.  Until  a  few  years 
ago  all  gears  of  this  type  were  molded.  The  limitations  of 
molded  gearing  are  analogous  to  those  which  would  be  expe- 
rienced if  a  journal  were  run  in  a  molded  bearing.  Just  as  the 
bearing  would  touch  the  shaft  only  in  spots,  so  molded  gears 
fail  to  give  the  intimate  contact  all  along  the  teeth  which  is 
necessary  to  secure  the  realization  of  true  helical  gear  action. 
It  is  obvious  that  if  the  teeth  touch  only  in  a  few  high  places, 
they  will  be  subjected  to  all  the  evils  of  shock,  stress  and  in- 
equality of  motion  which  it  is  desired  to  avoid.  If  the  gears 
are  particularly  well  molded,  some  mitigation  of  these  evils 
may  be  expected  when  they  become  well  worn,  but  the  initial 
wear  is  accompanied  by  a  departure  from  the  correct  tooth 
shape. 

For  slow  speeds  a  well-molded  helical  gear  is  no  better  than 
a  straight  gear  with  cut  teeth,  and  for  high  speeds  it  is  not  as 
good.  The  natural  smoothness  of  helical  action  does  no  more 
than  compensate  for  the  inaccuracies  of  tooth  form  and  spac- 
ing. The  modern  herringbone  gear  must  have  cut  teeth  if  its 
advantages  are  to  become  realized. 

One-  and  Two-piece  Types.  —  Cut  herringbone  gears  may  be 
broadly  divided  into  two  classes,  two-piece  and  one-piece  gears. 
The  difficulty  in  the  way  of  cutting  double  helical  teeth  in  a 
single  blank  gave  rise  to  the  two-piece  variety.  The  same 
methods  of  cutting  may  be  used  for  both  kinds.  The  disad- 
vantages of  the  two-piece  type  are  obvious.  There  is  the  expense 
of  two  complete  gears,  the  difficulty  of  assembling  so  that  the 


76  SPIRAL  GEARING 

gears  be  in  accurate  register  with  each  other,  and  the  necessity 
for  very  thorough  fastenings  if  they  are  to  perform  hard  service 
without  getting  out  of  register.  High  ratios  are  not  within  the 
scope  of  the  built-up  gear,  because  the  pinions  must  be  assembled 
on  a  separate  shaft  and  the  pitch  line  must  be  far  enough  from 
the  surface  of  the  shaft  to  allow  room  for  the  necessary  bolts 
or  rivets  used  in  fastening  the  two  portions  together.  The 
one-piece  pinion,  however,  may  be  cut  solid  with  its  shaft,  so 
that  its  pitch  diameter  need  be  but  very  little  larger  than  the 
latter. 

The  methods  of  cutting  helical  gears  may  be  divided  into  four 
classes:  (a)  milling  by  formed  disk  cutters;  (b)  milling  by  end- 
mills;  (c)  generating  by  shaping  or  planing  methods;  (d)  gen- 
erating by  hobs. 

Milling  by  Ordinary  Disk  Cutters.  —  Milling  by  formed  disk 
cutters  is  unsatisfactory,  because,  in  addition  to  the  usual  errors 
of  step-by-step  division,  there  is  the  difficulty  of  making  the 
cutters  to  the  normal  tooth  shape  with  sufficient  accuracy  to 
insure  correct  circumferential  shape  for  the  gears  cut.  This 
difficulty  is  increased  by  reason  of  the  fact  that  a  disk  cutter 
cannot  cut  its  own  shape  in  a  spiral  groove.  Let  it  be  noted 
that  the  cutters  must  be  formed  empirically,  that  their  number 
must  be  very  large  to  meet  the  requirements  of  a  general  gear 
business,  and  that  the  accuracy  of  each  gear  turned  out  depends 
on  the  combined  efforts  of  the  toolmaker  and  draftsman  who 
produced  the  cutter.  Worst  of  all,  two  different  cutters  must 
be  used  for  a  gear  and  pinion.  This  method  will  produce  in- 
different herringbone  gears  whether  they  are  built  up  with  teeth 
in  register  or  made  in  one  piece  with  staggered  teeth. 

Milling  by  End-mills.  —  The  use  of  end-mills  is  open  to  all 
the  objections  to  disk  cutters,  with  the  single  exception  that  the 
cutter  does  leave  a  fair  approximation  to  its  own  shape  in  the 
groove  which  it  cuts;  but  the  end-mill  has  a  number  of  dis- 
advantages peculiar  to  itself  which  render  it  even  less  efficient 
than  the  disk  cutter  for  general  work.  In  the  first  place  it  is  a 
small  tool  with  very  little  wearing  surface  and  no  capacity  for 
dissipating  the  heat  generated  at  its  cutting  edges.  The  great 


HERRINGBONE  GEARS  77 

variation  in  diameter  between  point  and  base  renders  it  difficult 
to  arrive  at  a  cutting  speed  which  will  satisfy  the  conditions  at 
both  ends  of  the  cut.  The  mills  quickly  become  clogged  with 
cuttings,  overheat  and  burn.  To  complete  one  fair-sized  gear 
by  the  end-mill  process  requires  quite  a  number  of  cutters.  This 
not  only  makes  the  expense  heavy,  but  must  necessarily  result 
in  an  inaccurate  gear. 

Every  cutter  used  must  be  formed  to  gage  and  hardened.  After 
being  hardened,  it  will  run  a  trifle  out  of  true,  in  most  cases, 
thereby  cutting  a  shape  different  from  that  for  which  it  was 
designed.  In  end-milled  gears  it  is  not  merely  a  case  of  getting 
accurate  conjugate  tooth  shapes  in  gear  and  pinion  made  with 
different  cutters,  but  the  teeth  in  a  single  gear  may  have  a  dozen 
different  shapes.  The  process  is  so  slow  that  it  cannot  compete 
with  other  methods,  quite  apart  from  the  doubtful  quality  of 
the  gears  produced. 

End-milled  herringbone  gears  are  usually  made  in  one  piece 
with  the  teeth  joined  at  the  center.  Since  the  cutter  is  shaped 
to  the  normal  pitch,  it  follows  that,  in  changing  over  from  right- 
to  left-hand  helix,  it  leaves  a  thick  wedge  in  the  center  of  the 
face  which  must  be  removed  by  a  subsequent  operation.  The 
teeth  of  end-milled  herringbone  gears  do  not  bear  over  the  center 
portion. 

Planing  the  Gear  Teeth.  —  Generating  processes  of  the  shap- 
ing and  planing  type,  while  successful  for  straight-cut  gears  of 
relatively  small  size,  are  not  used  to  any  extent  for  large  diam- 
eters or  heavy  pitches.  The  reason  for  this  may  be  found  in  the 
nature  of  the  processes.  The  gear  blank  is  required  to  make  a 
quick  angular  movement  after  each  stroke  of  the  cutting  tool 
and  to  come  to  rest  again  before  the  next  stroke.  Such  methods 
are  difficult  to  apply  to  large  gears  on  account  of  the  inertia  of 
the  gear  blank  and  its  support  and  the  consequent  difficulties  of 
controlling  the  short  intermittent  movements.  These  difficulties 
are  much  increased  when  such  methods  are  applied  to  cutting 
helical  teeth  because  the  blank  must  make  definite  and  rapid 
angular  movements  during  each  stroke  in  addition  to  the  mo- 
tion between  strokes. 


78  SPIRAL  GEARING 

The  Robbing  Process.  —  The  bobbing  process  as  applied  to 
straight-cut  gears  has  proved  very  successful,  and  it  is  not 
difficult  to  understand  why  this  process  has  sprung  into  prom- 
inence in  a  comparatively  short  time.  It  is  essentially  a  rational 
process.  The  shape  of  the  teeth  is  generated  from  spiral  hobs, 
the  threads  of  which  are  cut  to  a  plain  rack  section.  One  hob 
will  cut  any  gear  or  pinion  of  one  pitch.  This  feature  alone 
eliminates  a  great  many  errors  which  are  characteristic  of  gears 
produced  by  milling  methods.  The  hob  revolves  continuously 
while  cutting,  as  does  the  gear  blank.  The  feed  is  also  con- 
tinuous. There  are  no  cutting  and  return  strokes,  and  no 
intermittent  starting  and  stopping  of  gear  blanks,  as  in  other 
generating  processes.  These  features  do  not  necessarily  insure 
the  production  of  accurate  gears,  but  they  offer  greater  facilities 
to  the  designer  for  the  achievement  of  the  desired  result. 

The  hob  is  a  substantial  tool  with  plenty  of  wearing  and  cool- 
ing surface,  and  can  be  made  to  meet  the  demands  of  rapid 
production  and  to  last  for  a  long  time.  The  continuous  nature 
of  all  motions  used  in  hobbing  a  gear  blank  enables  this  process 
to  be  used  for  the  production  of  the  heaviest  gears.  The  limit 
to  the  size  of  a  hobbing  machine  is  set  by  the  dimensions  of  the 
largest  gears  which  are  required  in  sufficient  quantities  to  pay 
for  the  investment. 

Nevertheless,  there  are  some  defects  in  the  hobbing  process 
as  applied  to  the  production  of  straight-cut  spur  gears.  A  hob 
is  a  worm  thread,  and  as  such  must  have  a  spiral  angle  depend- 
ing on  the  relationship  between  the  pitch  of  the  thread  and  the 
diameter  of  the  hob.  A  straight-cut  gear  has  no  spiral  angle, 
hence  the  spiral  hob  must  be  inclined,  more  or  less,  to  bring 
the  cutters  in  line  with  the  tooth  spaces  to  be  cut.  In  order  to 
cut  correct  teeth,  the  axis  of  the  hob  should  be  perpendicular 
to  the  axis  of  the  gear  blank.  In  such  case  the  hob  will  generate 
involute  teeth  if  its  threads  are  cut  to  the  same  axial  section  as 
the  straight-sided  parent  rack  for  the  required  pitch.  Since 
the  hob  must  be  inclined  to  cut  a  spur  gear,  the  teeth  are  not 
generated  from  the  axial  or  rack  section,  but  from  a  diagonal 
section.  The  axial  pitch  of  a  hob  for  cutting  spur  gears  is  not 


HERRINGBONE  GEARS  79 

the  same  as  the  pitch  of  the  gears  which  it  cuts.  The  normal 
pitch  of  the  hob  threads  must  be  the  same  as  the  gear  pitch. 

Hobs  for  cutting  straight  spur  gears  are  usually  made  of  large 
diameter  to  reduce  the  spiral  angle  and  consequent  errors  of 
tooth  form  to  a  negligible  minimum.  As  a  natural  consequence, 
such  hobs  have  only  one  thread,  while  their  large  diameter  re- 
quires a  slow  speed  of  rotation  to  keep  the  cutting  speed  within 
proper  limits.  The  effect  of  this  is  that  the  blank  revolves  very 
slowly,  and  a  coarse  feed  must  be  used  to  keep  up  the  output. 

It  is  one  of  the  peculiarities  of  the  hob  action  that  only  one 
tooth  of  the  hob  puts  the  finishing  touch  to  the  bottom  of  a  tooth 
space  once  in  each  revolution  of  the  gear  blank.  If  the  feed  is 
coarse,  there  will  be  noticeable  feed  marks  and  roughness  in  the 
gear  teeth  produced.  A  coarse  feed  used  with  a  hob  of  large 
radius  throws  severe  stresses  on  the  hob  arbor  and  its  supports. 

The  necessity  of  a  swivel  motion  on  the  hob  slide,  to  enable 
straight  spurs  to  be  cut  with  hobs  of  varying  spiral  angle,  com- 
pels the  use  of  a  hob  drive  which  passes  through  the  pivot.  It 
is  almost  impossible  to  design  such  an  arrangement  without 
undesirable  restriction  in  the  dimensions  of  driving  gears  and 
shafts  combined  with  excessive  overhang  of  the  hob  arbor  in 
relation  to  its  supporting  slide.  The  general  lack  of  rigidity 
about  most  hobbing  machines  used  for  the  production  of  spur 
gears  is  traceable  to  the  above  causes.  Rational  critics  of  the 
hobbing  process  have  based  their  objections  on  these  features. 

The  hobbing  process  properly  applied  to  the  production  of 
herringbone  gears  has  none  of  the  disadvantages  incidental  to 
its  application  to  spur  gear  cutting,  which  have  been  shown  to 
lie  in  the  necessity  of  inclining  the  hob  axis.  Since  a  helical  gear 
and  a  hob  must  both  have  a  spiral  angle,  it  is  only  necessary  to 
make  the  thread  angle  of  the  hob  the  complement  to  the  corre- 
sponding angle  of  the  gear  teeth  to  secure  the  advantages  of 
perpendicular  fixed  axes.  This  principle  is  of  great  practical 
value.  Since  the  hob  axis  is  always  perpendicular  to  the  axis 
of  the  gear  blank,  it  follows  that  the  teeth  are  generated  from 
the  axial  and  true  rack  section  of  the  hob,  while  the  linear  pitch 
of  the  hob  is  the  same  as  the  circular  pitch  of  the  gear  which  it 


8o 


SPIRAL   GEARING 


cuts.  The  hob  axis  is  fixed  and  the  hob  can  be  supported  on 
a  rigid  slide  with  the  minimum  of  overhang.  There  is  no  re- 
striction to  the  size  and  strength  of  the  gears  and  shafts  used  to 
drive  the  hob. 

Wuest  Herringbone  Gears.  —  It  has  been  explained  that  the 
teeth  of  the  Wuest  gears  are  so  designed  that  those  on  the  right- 
and  left-hand  sides  of  the  gears  are  stepped  half  a  space  apart 
and  do  not  meet  at  a  common  apex  at  the  center  of  the  face, 
as  in  the  usual  type  of  herringbone  gear.  It  has  often  been 


Machinery 


Figs.  2  to  4.     Diagrams  showing  Tooth  Pressures  and  Angle 
Necessary  for  Continuity  of  Action 

argued  that  the  ordinary  herringbone  tooth  is  stronger  than  the 
Wuest  tooth,  because  the  latter  lacks  the  support  given  by  the 
junction  of  the  teeth  at  the  center.  This  argument  would  be 
sound  if  gear  teeth  were  ever  stressed  to  anywhere  near  their 
breaking  point.  It  has  been  found  in  practice  that  consider- 
ations of  wear  so  far  outweigh  those  of  mere  breaking  strength 
that  a  gear  which  is  designed  to  give  reasonable  service  will  carry 
anywhere  from  ten  to  twenty  times  the  working  load  without 
fracture.  A  point  of  vastly  greater  importance  is  that  the  stepped 
form  will  wear  more  evenly  under  extreme  loads  than  the  ordinary 
type.  The  reason  for  this  is  shown  in  Figs.  2  and  3.  The 
resultant  tooth  pressure  is  always  normal  to  the  teeth  and  tends 


HERRINGBONE  GEARS  8l 

to  bend  them  apart.  The  stepped  form  offers  a  uniform  re- 
sistance along  its  whole  length,  carrying  the  load  from  end  to 
end  (Fig.  2).  The  teeth  of  ordinary  herringbone  gears  tend  to 
separate  more  at  the  sides  than  near  the  supported  center, 
causing  the  load  to  be  concentrated  toward  the  center  (Fig.  3). 

Interchangeable  Herringbone  Gears.  —  Any  system  of  gear- 
ing, if  it  is  to  be  generally  applied,  must  be  interchangeable. 
The  variable  features  of  involute  spur  gear  teeth  are  limited  to 
the  pressure  angle,  addendum  and  dedendum.  In  a  herringbone 
gear  system,  we  must  have,  besides,  uniformity  of  spiral  angle 
and  relative  position  of  the  right-  and  left-hand  teeth. 

The  standards  which  have  been  adopted  for  Wuest  gears  are 
the  result  of  experience  gained  in  Europe  during  a  period  of 
years.  The  spiral  angle  of  the  teeth  is  about  23  degrees  with 
the  axis.  The  choice  of  this  angle  is  controlled  by  a  number  of 
considerations,  the  most  important  from  the  user's  standpoint 
being  that  the  angle  must  be  sufficient  to  allow  the  engagement 
of  successive  pinion  teeth  to  overlap  within  a  reasonable  face 
width.  Once  this  condition  is  satisfied,  there  is  no  advantage 
,in  an  increase  of  spiral  angle,  while  there  are  disadvantages  in 
the  use  of  steep  angles.  It  was  necessary,  before  choosing  a 
definite  spiral  angle,  to  determine  what  constitutes  a  reasonable 
face  width  for  this  class  of  gearing. 

Width  of  Face  and  Spiral  Angle.  —  Since  the  nature  of  the 
action  eliminates  shock,  it  follows  that  the  pitch  required  for 
given  conditions  will  be  much  finer  than  would  be  chosen  for 
spur  gears.  On  the  other  hand,  the  face  width  will  not  be  less, 
because  there  is  as  much  necessity  for  wearing  surface  with  one 
kind  of  tooth  as  with  the  other.  Spur  gears  are  usually  made 
with  a  face  width  equal  to  three  or  four  times  the  pitch.  Her- 
ringbone gears  may  conveniently  have  a  face  width  equal  to 
six  times  the  pitch,  not  because  the  width  of  this  type  actually 
is  greater,  but  because  the  pitch  is  proportionately  less. 

Starting  with  a  width  equal  to  six  times  the  pitch,  and  allow- 
ing a  width  equal  to  the  pitch  as  the  non-bearing  portion  at  the 
center,  there  remains  two  and  one-half  times  the  pitch  available 
for  the  teeth  on  each  side.  To  insure  continuity  of  engagement 


82 


SPIRAL  GEARING 


under  all  ordinary  conditions,  each  tooth  is  inclined  so  as  to  cover 
an  advance  equal  to  the  pitch  within  its  length.  The  angle  of 
23  degrees  satisfies  this  requirement  (see  Fig.  4).  There  are  a 
few  cases  where  an  angle  less  than  23  degrees  would  be  sufficient, 
while  a  steeper  angle  is  only  needed  if  the  available  face  width 
has  to  be  unduly  restricted.  Neither  of  these  extreme  condi- 
tions should  influence  the  choice  of  angle  for  an  interchangeable 
system  best  adapted  for  general  use. 

There  are  other  good  reasons  why  a  moderate  spiral  angle  is 
to  be  preferred.     In  all  spiral  gears  the  pressure  acts  in  a  direc- 


Machinery 


Fig.  5.     Diagrams  showing  Relation  between  Normal  Pressure,  Spiral 
Angle  and  Available  Normal  Tooth  Section 

tion  normal  to  the  teeth  and  is  the  resultant  of  the  tangential 
(driving)  and  axial  pressures.  The  normal  pressure  becomes 
greater  in  proportion  to  the  useful  driving  pressure  as  the  spiral 
angle  is  increased,  while  the  available  normal  tooth  section  be- 
comes less  (see  Fig.  5).  When  the  spiral  angle  is  considerably 
steeper  than  the  angle  of  repose  for  the  materials  in  contact, 
there  is  a  tendency  for  the  teeth  to  bind  with  a  wedge  action. 
Herringbone  gears  with  abnormally  steep  spiral  angles  show 
loss  of  efficiency  and  increased  wear  from  this  cause. 

Pressure  Angle  and  Tooth  Proportions.  —  The  pressure  angle 
which  has  been  adopted  for  standard  gears  is  20  degrees.     The 


HERRINGBONE  GEARS  83 

teeth  are  shorter  than  the  usual  standards,  as  indicated  by  the 
formulas  in  the  following.  These  standards  of  tooth  height  and 
pressure  angle  have  been  adopted  after  systematic  trials  and 
experience  extending  over  several  years  of  regular  manufacture. 
The  high  ratios  used  with  these  gears  call  for  an  average  pinion 
diameter  which  is  less  than  is  used  with  straight  spur  gears  for 
similar  duty.  The  teeth  are  generated  by  hobs,  and  the  short 
addendum  combined  with  large  pressure  angle  gives  satisfactory 
tooth  shapes,  without  undercutting  of  teeth,  for  small  pinions. 
Pinions  with  very  few  teeth  are  cut  on  the  well-known  system 
of  enlarged  addendum  which  is.  also  used  for  bevel  pinions. 
The  teeth  are  cut  to  diametral  pitch  standards,  measured  cir- 
cumferentially  the  same  as  in  ordinary  spur  gearing. 

Diametral  Pitch  of  Herringbone  Gears.  —  Many  designers 
find  it  difficult  to  regard  herringbone  gears  as  spur  gears.  They 
apply  the  same  methods  as  are  used  for  calculating  ordinary 
spiral  gears,  and  complicate  the  problem  to  an  unnecessary 
extent.  Spiral  gears  are  usually  employed  for  connecting  shafts 
which  are  not  parallel  to  each  other.  Under  these  conditions 
the  circumferential  pitch  of  gear  and  pinion  may  be  quite  differ- 
ent, but  the  normal  pitch  of  both  must  be  the  same. 

In  herringbone  gears,  if  the  spiral  angle  is  made  constant 
there  is  a  definite  and  fixed  relationship  between  the  normal  and 
the  circumferential  pitch.  This  is  the  case  with  Wuest  herring- 
bone gears.  It  is  a  great  convenience  to  discard  all  reference 
to  the  normal  pitch  and  treat  the  gears  just  like  spur  gears  on 
the  basis  of  the  circumferential  pitch.  When  this  is  once 
done,  it  makes  no  difference  whether  the  circular  or  diametral 
pitch  system  is  used.  It  is,  of  course,  necessary  for  the  gear 
cutter  to  set  his  calipers  to  the  normal  tooth  thickness,  and  if 
circular  cutters  or  inclined  hobs  are  used  they  must  be  designed 
for  the  normal  pitch  the  same  as  in  regular  spiral  gearing;  but 
the  designer  of  machinery  involving  the  use  of  these  gears  need 
not  be  troubled  with  any  such  complications. 

Wuest  herringbone  gears  are  cut  by  specially  constructed  hobs 
which  are  used  with  the  hob  axis  perpendicular  to  the  gear  axis. 
The  pitch  of  each  hob,  measured  along  the  axis  in  the  same  way 


84  SPIRAL  GEARING 

as  the  pitch  of  a  screw  is  measured,  is,  therefore,  the  same  as 
the  circumferential  pitch  of  the  gears  which  it  cuts. 

Pitch  Diameters  and  Center  Distances.  —  The  question  in 
regard  to  the  use  of  enlarged  pinions  is  not  so  easily  understood 
and  requires  a  clear  definition  of  what  constitutes  the  "  pitch 
diameter"  and  the  " pitch"  of  a  gear. 

If  the  center  distance  and  velocity  ratio  are  given,  then  the 
true  pitch  diameters  of  the  gear  and  pinion  are  fixed.  Now 
it  is  well  known  that  involute  gears  will  run  satisfactorily  when 
set  farther  apart  than  the  designed  center  distance.  In  other 
words,  the  center  distance  may  be  varied  to  a  limited  extent. 
This  variation  of  the  center  distance  does  not  effect  either  the 
number  of  teeth  or  the  velocity  ratio,  but  it  alters  the  pitch. 
The  foregoing  arguments  lead  to  the  curious  conclusion  that  the 
pitch  of  a  pair  of  involute  gears  has  no  definite  value,  but  de- 
pends on  the  center  distance  and  velocity  ratio.  Conversely,  if  we 
maintain  a  fixed  center  distance  and  ratio  for  a  given  pair  of  gears,  we 
can  cut  involute  teeth  in  various  ways  without  altering  the  pitch. 

For  instance,  if  we  require  a  small  pinion  to  mesh  with  a  large 
gear,  we  may  generate  the  teeth  to  standard  thickness  on  their 
true  pitch  diameters  or  we  may  enlarge  the  blank  diameter  of 
the  pinion  and  reduce  that  of  the  gear  by  a  corresponding  amount. 
The  teeth  will  be  generated  from  the  same  base  circles  in  each 
case,  and  the  true  pitch  diameters  and  pitch  will  be  the  same, 
but  the  shape  of  the  teeth  will  be  quite  different  in  the  two  cases. 
The  pinion  which  is  cut  on  standard  lines  will  probably  have 
badly  undercut  teeth  with  consequent  weakness  and  loss  of 
wearing  surface.  The  enlarged  pinion,  on  the  contrary,  will 
have  teeth  with  broad  bases  and  unimpaired  shape.  Since 
the  center  distance  and  velocity  ratio  have  not  been  altered,  the 
true  pitch  circles  and  the  pitch  remain  unchanged;  but  the 
change  in  outside  diameters  has  increased  the  addendum  of 
the  pinion  and  decreased  the  addendum  of  the  gear. 

There  is  nothing  new  in  this  method,  as  it  has  been  in  use  on 
bevel  and  worm  gears  for  many  years;  the  most  curious  thing 
about  it  is  that  it  continues  to  be  so  little  understood  by  the 
majority  of  gear  users. 


HERRINGBONE  GEARS  85 

An  enlarged  pinion  will  mesh  correctly  with  any  gear  in  its 
series,  whether  reduced  or  not,  but  if  the  gear  is  of  standard 
proportions  the  center  distance  will  be  greater  than  standard 
by  half  the  enlargement  of  the  pinion.  This  applies  to  all 
involute  gears  with  generated  teeth,  no  matter  whether  they 
are  hobbed,  shaped  or  planed.  When  this  method  is  applied 
to  herringbone  gears,  the  enlargement  or  reduction  of  the  blank 
is  left  entirely  out  of  consideration,  and  the  machine  is  set  to 
cut  the  correct  spiral  angle  on  the  true  pitch  circle.  Given  a 
proper  degree  of  accuracy  in  the  cutting  and  reasonable  care  in 
setting  up,  such  gears  are  perfectly  interchangeable,  bear  evenly 
from  end  to  end  and  do  not  jam.  There  is  no  question  of 
approximation.  These  methods  have  been  in  use  for  several 
years,  and  there  are  thousands  of  gears  cut  in  this  manner  giving 
satisfactory  service. 

Dimensions.  —  The  dimensions  proposed  for  an  interchange- 
able system  for  these  gears  are,  therefore,  as  follows : 

Tooth  shape Involute 

Pressure  angle 20  degrees 

Spiral  angle 23  degrees 

-P,.,  ,    ,.              f           ,,                   v       Number  of  teeth 
Pitch  diameter  (20  teeth  and  over)  =  — — 

•m     !   j.             f          ,,                  N      Number  of  teeth  + 1 .6 
Blank  diameter  (20  teeth  and  over)  =  — -— 

x       O.CK  X  No.  of   teeth  •+•  i 
Pitch  diameter  (under  20  teeth)  =  — — — ! — 

-m     i   j-              /     j               41  \      o-95  X  No.  of  teeth  +  2.6 
Blank  diameter  (under  20  teeth)  =  -  — — — — 

For  all  herringbone  gears,  irrespective  of  number  of  teeth : 

0.8 


Addendum  . .  . 
Dedendum  . . . 
Full  depth  . . . 
Working  depth 


D.P. 

i.o 
D.P. 

1.8 
D.P. 

1.6 
D.P. 


86  SPIRAL  GEARING 

Standard  face  width  for  gears  with  pinions  of  not  less  than 
25  teeth,  6  times  circular  pitch;  face  widths  for  high  ratio  gears 
with  small  pinions,  6  to  12  times  circular  pitch. 

When  a  pinion  of  less  than  20  teeth  is  used  with  a  standard 
gear,  the  center  distance  must  be  slightly  increased  to  suit  the 
enlargement  of  the  pinion.  If  it  is  desired  to  keep  the  center 
distance  to  the  standard  dimensions,  the  gear  diameter  may  be 
reduced  by  the  amount  of  the  enlargement  given  to  the  pinion. 
For  example,  if  a  pinion  of  10  teeth,  5  diametral  pitch  is  to  mesh 
with  a  gear  of  90  teeth  at  lo-inch  centers,  then: 

-nv   u   J*  f      •    •  °-9S  X  10  +  I  .      . 

Pitch  diameter  of  pinion  =  -  =2.1  inches. 

0 

Enlargement  over  standard  pinion  =  o.i  inch. 
Pitch  diameter  of  standard  gear  =  —  =  18.0  inches. 
Reduced  pitch  diameter  of  gear  =  18.0  —  o.i  =  17.9  inches. 

Center  distance  =  ^^ —  =  10  inches. 

2 

Strength  of  Herringbone  Gears.  —  In  these  gears  the  teeth 
need  not  have  the  same  breaking  strength  as  with  spur  gears 
because  they  have  not  to  combat  the  heavy  and  indeterminate 
stresses  which  arise  from  inequalities  of  angular  velocity.  On 
the  other  hand  it  is  necessary  to  provide  against  rapid  wear. 
By  using  a  finer  pitch,  each  tooth  has  less  individual  wearing 
surface,  but  this  is  more  than  compensated  for  by  the  larger 
number  of  teeth  in  simultaneous  contact  than  is  the  case  with 
gears  of  equal  diameters  but  coarser  pitch. 

In  high  ratio  gears,  using  pinions  of  exceptionally  small 
diameter,  the  pitch  is  finer  than  for  ordinary  ratios,  but  the  face 
width  is  extended  to  give  the  proper  wearing  surface. 

The  important  factor  in  determining  the  proportions  of  the 
teeth  is  the  relationship  between  pitch  line  velocity  and  the  per- 
missible specific  tooth  pressure;  in  other  words,  the  total  tooth 
pressure  divided  by  the  area  of  all  the  available  simultaneous 
contact  along  the  teeth.  Theoretically,  this  contact  has  no 
area  since  it  should  consist  of  lines  without  breadth.  Actually, 


HERRINGBONE  GEARS  87 

an  area  exists,  due  to  the  elastic  compression  of  the  teeth  in 
contact,  in  a  similar  way  in  which  an  area  of  contact  exists 
between  a  car  wheel  and  a  rail.  The  area  of  contact  is  inde- 
terminate, but  the  specific  tooth  pressure  is  proportional  to  the 
driving  stress  on  the  teeth. 

Horsepower  Transmitted.  —  In  order  to  obtain  a  simple  rule 
for  finding  the  proper  dimensions,  the  results, of  experience  in 
the  matter  of  safe  working  loads  under  given  conditions  have 
been  reduced  to  a  relationship  between  pitch  line  velocity  and 
the  shearing  stress  on  the  pitch  line  thickness  of  an  imaginary 
straight  tooth,  assuming  only  one  tooth  in  engagement  at  a 
time.  The  shearing  stress  is  a  measure  of  the  specific  tooth 
pressure,  and  the  relationship  referred  to  affords  a  convenient 
means  of  arriving  at  reliable  dimensions.  The  curves,  Fig.  6, 
give  values 'of  shearing  stress  K  in  pounds  per  square  inch  on 
pitch  line  section  of  an  imaginary  single  tooth  for  corresponding 
pitch  line  velocities  V  in  feet  per  minute.  The  values  are  en- 
tirely empirical,  but  they  are  based  on  the  results  of  extended 
experience,  and  lead  to  dimensions  which  are  safe  and  reliable. 
Different  curves  are  given  for  different  materials,  and  it  is  neces- 
sary to  use  that  curve  which  corresponds  to  the  lowest  grade 
material  of  the  combination.  A  table  is  also  provided  in  which 
approximate  values,  taken  from  the  diagram,  are  given.  The 
dimensions  of  gears  can  be  derived  from  the  curves  in  the  follow- 
ing manner.  In  the  formulas: 

H.P.  =  horsepower  transmitted; 
N  =  revolutions  per  minute; 
D  =  pitch  diameter  in  inches; 

p  =  circular  pitch  in  inches; 

F  =  total  width  of  face  in  inches; 

V  =  pitch  line  velocity  in  feet  per  minute; 
W  =  total  tooth  pressure  at  pitch  line  in  pounds; 
K  =  stress  factor  as  obtained  from  table. 


H.P.  X  33,000  _  pFK 
V  2.5 


88 


SPIRAL  GEARING 


HONI  3avnos  a3d  sasmod  NI  y\ 
;i88S888 


3-iaissii/\iav 


8 

IS 

HONI  3uvnos 


888888888 

00*^-^*""1 

NI  >j  8S3u±s 


HERRINGBONE  GEARS 


In  gears  with  moderate  ratio  (not  exceeding  i  to  6),  and  face 
width  equivalent  to  6  times  the  circular  pitch,  make: 


or 


p  = 


W 


For  higher  ratios  make  F 
up  to  a  maximum  of  F  =  10  p. 
is  found  from: 


2.4  K 

Rp  (where  R  =  ratio  of  gears) 
The  circular  pitch  for  high  ratios 


When  the  face  width  is  equivalent  to  8  times  the  circular 
pitch,  W  =  3.2  p2K,  and  when  the  face  width  is  equivalent  to 
10  times  the  circular  pitch,  W  =  4  p2K. 

Table  of  Safe  Shearing  Stresses  K,  in  Pounds  per  Square  Inch,  for 
Herringbone  Gears 


Factor  K  for 

Velocitv  in  Feet 

per  Minute 

Brass 

Cast 
Iron 

Gun 

Metal 

Phosphor 
Bronze 

Steel 
Castings 

High-car- 
bon Steel 
Forgings 

IOO 

600 

800 

IOOO 

1150 

1325 

1800 

200 

575 

750 

95° 

I  IOO 

1275 

1750 

300 

550 

700 

900 

1060 

1250 

1700 

400 

525 

675 

860 

1030 

I20O 

1660 

500 

500 

650 

830 

IOOO 

H75 

1630 

600 

475 

625 

800 

975 

1150 

1600 

800 

425 

575 

75o 

925 

I  IOO 

1550 

IOOO 

400 

525 

700 

875 

1050 

1500 

I2OO 

380 

500 

650 

825 

IOOO 

1450 

1500 

360 

475 

600 

775 

925 

1350 

I800 

350 

45o 

550 

725 

875 

1275 

2100 

340 

425 

525 

675 

825 

I2OO 

24OO 

325 

400 

500 

650 

775 

H25 

3000 

300 

375 

475 

600 

700 

1050 

For  ordinary  service  it  is  safe  to  use  pitch  line  velocities  between 
looo  and  2000  feet  per  minute,  with  1500  feet  as  a  fair  aver- 
age. If  the  pinion  is  to  be  fixed  to  a  motor  shaft  without  ex- 
ternal support,  the  diameter  must  be  greater  than  when  it  can 
be  supported  on  both  sides.  Cast  iron  is  preferable  to  cast 
steel  for  gears  of  large  diameters  and  moderate  power,  but  the 
latter  will  be  found  more  economical  for  high  tooth  pressures. 


SPIRAL  GEARING 


Pinions  are  usually  made  from  steel  forgings  of  0.40  to  0.50 
per  cent  carbon.  Soft  pinions  should  never  be  used  for  her- 
ringbone gears. 

There  are  two  special  cases  where  the  ordinary  methods  of 
calculation  should  not  be  used.  Rolling-mill  gears  are  sub- 
jected to  stresses  which  are  so  far  in  excess  of  the  average  work- 
ing load  that  it  is  necessary  to  consider  carefully  the  strength 
of  the  teeth  in  regard  to  possible  overloads.  Extra  high  ve- 
locity gears,  such  as  are  used  for  steam  turbines,  require  addi- 
tional wearing  surface  and  are  characterized  by  extreme  width 
of  face  combined  with  abnormally  fine  pitch. 

The  following  is  a  typical  instance  of  the  range  of  choice  in 
dimensions:  A  pump  which  requires  150  horsepower  at  50  rev- 
olutions per  minute  is  to  be  driven  from  a  motor  at  500  revolu- 
tions per  minute;  the  shaft  end  is  4^  inches  in  diameter.  If 
the  shaft  is  unsupported,  it  is  not  desirable  to  use  a  pinion  of 
less  than  10  inches  diameter.  If  the  shaft  is  extended  to  a  third 
bearing  a  7|-inch  pinion  can  be  used.  If  the  pinion  is  cut  solid 
on  its  shaft  and  coupled  to  the  motor,  its  diameter  can  be  re- 
duced to  5  inches.  The  three  arrangements  work  out  as  fol- 
lows: 


Material 

Diametral 

Face 

for  Gear 

V 

W 

K 

Pitch 

Width 

i.   10    in.  and  TOO  in  — 

Cast  iron 

1300 

3800 

500 

2 

pH 

2.     7H  in.  and  75  in  

Cast  iron 

975 

5100 

530 

2 

12 

3.     7H  in.  and  75  in  

Cast  steel 

975 

5100 

1060 

2H 

7M 

4.     5     in.  and  50  in  

Cast  steel 

650 

7600 

1150 

2^ 

I2H 

Any  of  the  above  gears  will  do  the  work  satisfactorily;  (3)  is 
the  most  economical,  but  (2)  or  (4)  would  make  the  least  noise. 
If  a  gear  case  is  to  be  provided,  then  (4)  will  give  the  most 
economical  combination. 

The  foregoing  data  can  be  used  for  finding  the  required  di- 
mensions of  herringbone  gears  for  all  general  applications.  In 
most  cases  it  is  sufficient  to  calculate  the  tooth  pressure  from 
the  average  working  load.  When  the  maximum  load  is  very 
far  in  excess  of  the  average,  it  is  usual  to  take  a  mean  value 


HERRINGBONE  GEARS  91 

between  the  two.  Gears  for  electric  mine  hoists  and  single- throw 
pumps  fall  within  this  category.  Machine  tools,  when  driven 
from  variable-speed  motors,  are  required  to  perform  maximum 
duty  at  minimum  speed  only  for  short  periods  at  long  intervals. 
It  is  sufficient  when  designing  gears  for  a  drive  of  this  kind  to 
reckon  with  the  rated  output  of  the  motor  at  the  mean  between 
its  maximum  and  minimum  speed. 

General  Points  in  Design.  —  The  usual  conditions  met  with 
in  designing  any  form  of  gear  drive  are  that  power  is  to  be  trans- 
mitted from  a  motor  spindle  or  other  prime  mover  running  at 
high  speed  to  a  machine  which  is  required  to  operate  at  a  con- 
siderably reduced  speed.  In  selecting  a  pair  of  herringbone 
gears  (as  in  the  case  of  any  other  form  of  gearing)  the  designer 
naturally  selects  the  smallest  size  of  pinion  which  can  be  con- 
veniently adapted  to  the  service  for  which  it  is  desired.  By  so 
doing,  the  size  of  the  gear  is  reduced  so  that  the  least  possible 
amount  of  space  is  required.  Another  reason  for  selecting  a 
small  pinion  and  corresponding  gear  to  give  the  required  speed 
reduction  is  to  reduce  the  pitch  line  velocity  to  a  minimum,  as 
a  higher  velocity  means  increased  wear  and  objectionable  noise. 
Both  the  pinion  and  gear  have  the  same  pitch  line  velocity,  and, 
consequently,  the  same  tooth  strength.  In  the  design  of  her- 
ringbone gears  a  pinion  with  21  teeth  will  be  found  satisfactory 
for  average  conditions,  although  it  is  possible  to  have  a  pinion 
with  a  smaller  number  of  teeth,  pinions  with  as  few  as  13  teeth 
having  been  used  with  satisfactory  results.  Where  such  small- 
sized  pinions  are  used,  however,  they  are  made  solid  on  the 
shaft. 

Tables  of  Horsepower  Transmitted.  —  The  accompanying 
tables  give  the  horsepower  transmitted  by  herringbone  gearing 
for  various  pitches  in  cast  iron  and  cast  steel,  with  pitch  line 
velocities  ranging  from  400  to  2000  feet  per  minute.  These 
tables  are  used  for  determining  the  gear  which  is  necessary  for 
transmitting  a  given  power  at  a  specified  speed,  and  are  based 
on  the  formulas  already  given  for  calculating  the  horsepower 
transmitted.  It  is  customary  to  use  a  pinion  of  tougher  material 
than  the  gear,  owing  to  the  greater  wear  to  which  it  is  exposed, 


SPIRAL  GEARING 


Horsepower  Transmitted  by  Herringbone  Gears 


Velocity  at  Pitch 
Circle,  Feet  per 
Minute 

i^  Diametral  Pitch,  21  Teeth, 
14-inch  Pitch  Diameter 

\%  Diametral  Pitch,  21  Teeth, 
12-inch  Pitch  Diameter 

a  « 
|| 

i6-inch  Face 

20-inch  Face 

II 

f  ^ 

14  ^-in  Face 

i8-inch  Face 

Cast 
Iron 

Steel 
Cast- 
ing 

Cast 
Iron 

Steel 
Cast- 
ing 

Cast 
Iron 

Steel 
Cast- 
ing 

Cast 
Iron 

Steel 
Cast- 
ing 

400 
600 
800 
1000 
1200 

1400 
1500 
1600 

1800 

2000 

no 
160 
220 
270 

33° 
380 
410 

435 
490 

540 

110 
152 
I87 

215 
244 
270 
282 
292 

333 

199 

280 
358 
426 
488 
540 
564 
585 
640 
67I 

137 
190 

234 
269 

305 
338 
343 
366 
388 
406 

249 
350 
447 
534 
610 

675 

707 

732 
800 
838 

127 
190 
254 

382 
445 

477 

572 
636 

86 
118 
146 
166 
190 

2IO 
2I9 
228 
241 
258 

155 
218 

279 
331 
378 
420 
438 
455 
496 
522 

106 
146 
180 
206 
234 
261 
272 
282 
300 
320 

192 
271 
345 
413 
470 

545 
562 
610 
710 

Velocity  at  Pitch 
Circle,  Feet  per 
Minute 

2  Diametral  Pitch,  21  Teeth, 
io^-inch  Pitch  Diameter 

zYi  Diametral  Pitch,  21  Teeth, 
8.4-inch  Pitch  Diameter 

11 
11 

12^-inch  Face 

15^-inch  Face 

11 

lo-inch  Face 

I2l/i-inch  Face 

Cast 
Iron 

Steel 
Cast- 
ing 

Cast 
Iron 

Steel 
Cast- 
ing 

Cast 
Iron 

Steel 
Cast- 
ing 

Cast 
Iron 

Steel 
Cast- 
ing 

400 
600 
800 
1000 
1200 
1400 
I5OO 
I6OO 
1800 
2OOO 

145 
218 
291 
364 

437 

546 

655 
730 

64 
89 
109 
125 
143 
158 
166 
171 
182 
195 

116 
164 

207 
250 
285 
316 
329 
342 
375 
392 

79 
in 
136 
154 
178 
196 
206 
213 
226 
242 

144 
204 
259 
309 
355 
393 
410 

425 
465 
488 

182 
273 
364 
455 
546 
637 
683 

729 
820 
910 

41 
57 
70 
80 

91 
101 

106 
109 
1x6 

122 

74 
105 
134 
1  60 

185 
2O2 
211 
2  2O 
24O 
251 

51 
71 
87 
IOO 

114 
126 
132 
137 
145 
152 

93 
130 

167 
200 

228 

253 
264 

274 
300 

Velocity  at  Pitch 
Circle,  Feet  per 
Minute 

3  Diametral  Pitch,  21  Teeth, 
7-inch  Pitch  Diameter 

3H  Diametral  Pitch,  21  Teeth 
6-inch  Pitch  Diameter 

I* 

&l 

S^-inch  Face 

io>i-inch  Face 

'•£  c 

1! 

&  & 

7^4-inch  Face 

9-inch  Face 

Cast 
Iron 

Steel 
Cast- 
ing 

Cast 
Iron 

Steel 
Cast- 
ing 

Cast 
Iron 

Steel 
Cast- 
ing 

Cast 
Iron 

Steel 
Cast- 
ing 

400 
600 
800 

IOOO 
I2OO 
I4OO 
1500 
1600 
I800 
2OOO 

2x8 

327 

436 

546 

655 
765 

820 

875 
983 

1090 

28 
39 
48 
55 
63 
69 
72 

75 
80 
84 

51 
72 
92 
no 

126 
139 

146 

165 

36 
50 
61 
70 
80 
89 

93 
96 

102 
107 

65 
92 
117 
140 
160 
177 
184 
192 

210 
2  2O 

255 
382 

Sio 
637 
765 
892 

955 

1020 
1150 
1275 

21 
29 
36 
42 

47 
52 
55 
57 
61 

65 

39 
55 
70 

83 
95 
105 

IIO 

114 
123 

26 
36 
45 
52 
58 
65 
68 

76 
81 

48 
68 
87 
103 
118 
130 
136 
141 

162 

HERRINGBONE  GEARS 


93 


and  the  curves  in  Fig.  6  were  used  in  calculating  the  tables, 
for  obtaining  the  relative  toughness  of  the  different  metals 
which  are  ordinarily  used. 

The  tables  give  the  horsepower  transmitted  by  herringbone 
gears  in  which  the  pinion  has  21  teeth,  and  the  width  of  face 
corresponds  to  8  and  10  times  the  circular  pitch.  To  find  the 
horsepower  for  any  other  number  of  teeth,  ascertain  the  pitch 
line  velocity,  and  under  the  given  diametral  pitch,  find  the  horse- 
power corresponding  to  this  velocity.  To  find  the  horsepower 
transmitted  by  a  brass  gear,  multiply  that  found  for  a  cast-iron 

Horsepower  Transmitted  by  Herringbone  Gears 


JL 

oT  ^ 
•^Tj   C 

4  Diametral  Pitch, 
21  Teeth,  554-inch 
Pitch  Diameter 

5  Diametral  Pitch, 
21  Teeth,  4.2-inch 
Pitch  Diameter 

6  Diametral  Pitch, 
21  Teeth,  3^-inch 
Pitch  Diameter 

6M-inch 

7%-inch 

5-inch 

6H-inch 

4-inch 

5H-inch 

"3  J-,  fe 

S 

Face 

Face 

^ 

Face 

Face 

it 

Face 

Face 

^"  £ 

OH 

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P< 

PH 

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r  T 

S.C. 

CJ 

S.C. 

<£ 

r,  T 

RC 

C.I 

sr 

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S.C. 

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s  c 

16 

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13 

400 

292 

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IO 

13 

24 

436 

9 

17 

600 

437 

22 

41 

27 

51 

545 

14 

27 

18 

34 

655 

IO 

18 

13 

24 

800 

584 

27 

S3 

34 

66 

725 

17 

34 

21 

43 

873 

12 

23 

16 

30 

IOOO 

730 

31 

62 

39 

77 

910 

20 

40 

25 

50 

1090 

14 

27 

18 

35 

I2OO 

36 

71 

44 

88 

1090 

23 

46 

29 

58 

1310 

IS 

31 

20 

I4OO 

IO2O 

40 

80 

50 

99 

1270 

25 

51 

31 

64 

1525 

17 

35 

22 

46 

1500 

1090 

41 

83 

103 

1360 

26 

53 

32 

66 

1635 

18 

36 

24 

47 

I6OO 

1160 

43 

86 

53 

107 

145° 

27 

55 

34 

69 

1745 

19 

37 

25 

49 

I800 

1310 

46 

93 

57 

116 

1630 

29 

59 

36 

74 

1960 

20 

40 

26 

53 

2OOO 

1460 

49 

99 

61 

122 

1820 

31 

63 

39 

79 

2180 

21 

42 

28 

55 

gear  by  0.8;  for  gun  metal,  multiply  by  1.25;  and  for  phosphor- 
bronze,  by  1.63.  For  high-carbon  steel  forgings,  multiply  the 
horsepower  transmitted  by  steel-casting  gears  by  1.45. 

To  illustrate  the  method  of  using  the  tables,  it  is  assumed 
that  a  herringbone  gear  drive  is  required  to  transmit  100  horse- 
power from  a  motor  running  at  650  revolutions  per  minute  with 
a  speed  reduction  of  12  to  i.  In  the  table,  under  2|  pitch, 
lo-inch  face  and  opposite  637  revolutions  per  minute,  101  horse- 
power is  found  to  be  the  capacity  of  a  cast-iron  gear  running  at 
a  pitch  line  velocity  of  1400  feet  per  minute.  This  would 

necessitate  — —  =  100.8  inches  pitch  diameter  for  the  gear 


94  SPIRAL  GEARING 

to  mesh  with  a  2i-tooth,  2 ^-diametral  pitch  pinion.  The  pitch 
line  velocity  of  a  zy-tooth  pinion  would  be  1150  (or,  say  1200) 
feet  per  minute,  and  in  the  i2oo-foot  line,  it  is  seen  that  a 
gear  running  at  this  speed  would  transmit  91  horsepower. 
This  means  that  a  wider  face  gear  must  be  used,  if  a  ly-tooth 
pinion  is  to  find  successful  application,  the  width  of  face  required 

being =  n  inches.     The  pinion  in  either  case  should 

9i 

be  made  of  cast  steel  or  preferably  of  forged  steel,  to  compen- 
sate for  the  greater  wear  to  which  it  is  exposed. 

If  it  is  required  to  find  the  horsepower  transmitted  by  a  2-pitch 
cast-steel  gear,  having  a  pitch  diameter  of  90  inches  by  zo-inch 
face,  running  at  50  revolutions  per  minute,  we  find  that  the 
pitch  line  velocity  is  1180  feet  per  minute.  Opposite  1200  in 
the  table,  under  2  diametral  pitch  and  i2|-inch  face,  we  find 
285  horsepower,  while  the  capacity  of  a  gear  of  lo-inch  face 

would  be =  228  horsepower.     Although  the  tables  are 

calculated  for  21 -tooth  pinions,  they  are  universal  in  their 
application,  it  being  merely  necessary  to  find  the  corresponding 
pitch  line  velocity  for  any  number  of  teeth. 

Summary  of  Salient  Features.  —  Before  describing  some 
special  applications  of  herringbone  gears  to  the  needs  of  various 
industries  and  machines,  it  may  be  well  to  summarize  their 
salient  features.  The  smooth  and  continuous  action  is  virtu- 
ally independent  of  the  diameter  or  number  of  teeth  in  the 
pinion.  Extremely  high  ratios  of  reduction  can  be  used  with- 
out fear  of  uneven  driving  or  undue  wear  and  without  need  for 
unwieldy  gear  diameters  which  would  be  disproportionate  to  the 
general  design.  High  ratio  gears  of  this  type  transmit  power 
with  practically  no  more  loss  than  low  ratio  gears.  They  are 
far  more  efficient  than  belts,  ropes,  worm-gears  or  compound 
trains  of  spur  gearing,  while  their  adoption  results  in  a  whole- 
sale reduction  of  countershafts  and  bearings,  which  reduces  the 
power  consumption  and  running  costs  to  a  remarkable  degree. 

There  are  many  instances  where  spur  gears  cannot  be  used 
because  the  vibration  which  they  set  up  has  a  detrimental  effect 


HERRINGBONE  GEARS  95 

on  the  driven  machine  or  its  product.  The  inconvenience  of  a 
cumbersome  system  of  belts  or  ropes  has  usually  to  be  endured 
in  such  cases,  but  it  is  not  too  much  to  say  that  the  requirements 
of  almost  all  of  these  conditions  are  fully  satisfied  by  this  type 
of  herringbone  gears. 

The  application  of  spur  gears  has  been  much  restricted  by 
the  noise  which  they  make  when  run  at  high  velocities.  The 
use  of  rawhide  or  other  soft  materials  has  proved  successful 
for  comparatively  light  work,  but  is  not  adapted  to  low  speeds 
and  heavy  service.  It  should  be  noted  that  the  use  of  soft 
pinions,  while  mitigating  the  nuisance  of  excessive  noise,  does 
not  reduce  vibration  or  unevenness  of  motion.  There  is  a  limit 
to  the  pitch  line  velocities  at  which  spur  gears  can  be  operated 
beyond  which  it  is  unsafe  to  use  them.  This  limit  is  far  below 
the  minimum  velocities  which  can  be  used  in  connection  with 
steam  turbines  of  economical  design  and  high  power.  Accurate 
herringbone  gears  operate  quite  smoothly  at  velocities  which 
are  impossible  for  other  types.  This  feature  would  appear  to 
reserve  for  them  a  field  of  application  which  has  great  possi- 
bilities and  is  likely  to  cause  some  great  changes  in  the  standard 
practice  of  today. 

Application  to  Steam  Turbines. — Direct-connected  steam  tur- 
bines for  marine  propulsion  have  been  only  partially  successful 
in  a  very  limited  field.  It  is  only  when  the  power  required  is 
very  great  and  the  speed  of  the  vessel  unusually  high  that  the 
direct-connected  turbine  can  be  applied,  and  even  then  the  ap- 
plication does  not  do  full  justice  to  either  turbine  or  propeller, 
while  the  first  cost  is  much  higher  than  it  need  be.  The  use  of 
direct-coupled  turbines  is  confined  to  fast  ocean  liners  and  war- 
ships. Ordinary  vessels  of  commerce  cannot  be  adapted  to  tur- 
bine power  in  this  form.  Mr.  Parsons,  of  steam  turbine  fame, 
attacked  the  problem  of  applying  the  turbine  to  an  ordinary 
freight  steamer  of  moderate  power.  To  this  end  he  purchased 
the  S.  S.  Vespasian,  a  modern  tramp,  with  triple-expansion 
engines  of  about  1000  H.P.  and  a  speed  of  u  knots,  with  pro- 
peller running  at  75  R.P.M.  As  a  preliminary  to  the  installation 
of  geared  turbines  on  this  vessel,  the  original  engines  were  over- 


96  SPIRAL  GEARING 

hauled,  and  a  series  of  coal  consumption  trials  made  under 
regular  sea-going  conditions. 

The  engines  were  then  removed  and  for  them  were  substituted 
a  pair  of  steam  turbines  connected  to  the  propeller  by  herring- 
bone gears.  Each  turbine  develops  about  500  H.P.  at  1500 
R.P.M.  The  propeller  runs  at  the  original  speed  of  75  R.P.M. 
Each  turbine  is  coupled  to  a  herringbone  pinion  with  teeth  cut 
solid  on  a  shaft  of  soft-grade  chrome-nickel  steel.  The  two 
pinions  mesh  with  rolled-steel  gear  rings  mounted  on  a  cast-iron 
spider  which  is  keyed  to  the  propeller  shaft.  The  whole  gear 
system  is  enclosed  in  a  case,  and  the  teeth  are  kept  lubricated 
by  oil  jets.  The  great  width  of  the  pinions  in  proportion  to  their 
diameter  made  it  necessary  to  provide  room  for  bearings  be- 
tween the  right-  and  left-hand  teeth.  The  proportions  of  this, 
remarkable  gear  unit  are  as  follows:  Pinions,  20  teeth;  gear, 
398  teeth,  4  diametral  pitch;  teeth  of  involute  form,  2o-degree 
pressure  angle,  23-degree  spiral  angle;  over-all  face  width,  34 
inches,  including  10  inches  space  for  bearing;  actual  face  width, 
24  inches;  ratio  of  reduction,  19.9  to  i. 

When  this  gear  had  been  running  in  regular  voyages  for  more 
than  a  year  and  had  covered  over  20,000  miles,  the  results  that 
had  been  obtained  proved  to  have  been  interesting  and  satis- 
factory. The  efficiency  of  the  gear  was  fully  98  per  cent,  in- 
cluding the  losses  in  the  bearings  on  the  gear  case.  The  geared 
turbine  showed  a  sustained  all-around  saving  in  fuel  consumption 
of  more  than  25  per  cent  over  the  original  engines.  The  wear 
on  the  teeth  was  negligible  after  20,000  miles,  being  only  0.002 
inch  at  the  pitch  lines  of  the  pinions. 

Herringbone  Gears  for  Machine  Tools.  —  The  field  for  accu- 
rate herringbone  gears  in  connection  with  machine  tool  driving 
is  very  extensive.  For  individual  motor  drives  this  gear  gives 
a  positive  transmission  which  is  free  from  vibration  and  less 
noisy  than  so-called  silent  chains  or  rawhide  pinions,  while  there 
is  no  trouble  from  slipping  belts  or  slack  chains;  but  the  real 
advantage  of  these  gears  lies  in  the  better  finish  that  can  be 
obtained  when  they  are  used  for  the  entire  main  transmission, 
and  in  the  higher  output  combined  with  reduced  maintenance 


HERRINGBONE  GEARS  97 

which  they  give  to  heavy  machine  tools.  Chatter  is  eliminated. 
Even  wheels  of  grinding  machines  have  been  successfully  driven 
through  herringbone  gears.  Reversing  gears  for  heavy  planers 
are  a  revelation  to  those  familiar  only  with  the  ordinary  spur 
drive. 

Geared  Hydraulic  Turbines.  —  The  speed  of  hydraulic  tur- 
bines is  controlled  by  the  available  head  of  water  supplied  to 
them.  The  greater  number  of  turbines  are  required  to  operate 
under  low  heads  and  must  run  at  slow  speed.  Hydroelectric 
power  has  usually  to  be  transmitted  to  a  considerable  distance 
and  is  produced  in  the  form  of  alternating  current  of  definite 
periodicity.  The  speed  of  the  turbines  may  be  as  low  as  50  rev- 
olutions per  minute  or  even  less.  A  large  direct-coupled  alter- 
nator for  this  speed  is  an  expensive  proposition. 

Herringbone  gears  can  be  used  to  speed  up  from  the  slow- 
running  turbines  to  generators  of  normal  design,  speed  and 
efficiency.  The  smooth  action  of  these  gears  is  unimpaired 
when  the  wheel  drives  the  pinion,  and  high  ratios  of  speed  in- 
crease can  be  obtained  from  them  without  noise  and  with  less 
loss  than  direct-coupled  units  will  give. 

Rolling  Mills  and  Rod  Mills.  —  There  are  two  advantages  in 
the  use  of  accurate  herringbone  gears  for  this  class  of  work. 
The  absence  of  shock  in  transmission  renders  breakages  much 
less  frequent  than  with  cut  spur  or  molded  helical  gears.  The 
even  transmission  and  entire  elimination  of  vibration  allows 
the  finishing  rolls  to  be  gear-driven  for  the  finest  work  without 
showing  gear  marks  on  the  finished  product.  Herringbone- 
geared  mills  run  with  very  little  noise.  This  may  be  of  less 
consequence  in  rolling  mills  than  in  most  other  applications,  but 
it  is  an  improvement.  Rod  mills,  with  their  quantities  of  high- 
speed gearing,  can  be  very  advantageously  equipped  with  her- 
ringbone gears  and  pinions.  4 


CHAPTER  IV 

METHODS  FOR  FORMING  THE  TEETH  OF  SPIRAL  AND 
HERRINGBONE  GEARS  AND  WORMS. 

SPIRAL,  herringbone  and  worm  gearing  are  all  radically  differ- 
ent in  their  action.  The  first  two  forms,  however,  and  the 
worm  member  of  the  third,  are  identical  so  far  as  the  principles 
governing  the  forming  of  their  teeth  are  concerned,  and  they 
may,  therefore,  be  considered  together  in  this  chapter. 

Principal  Methods  Used.  —  Almost  as  great  a  variety  of 
methods  of  cutting  teeth  are  possible  for  helical  as  for  spur  gears. 
Commercially,  however,  the  two  important  principles  are  the 
formed  tool  and  the  molding-generating  methods.  The  templet, 
odontographic  and  describing-generating  methods  of  cutting 
gear  teeth  (in  each  of  which  the  outline  is  worked  out  by  the  point 
of  a  tool,  suitably  constrained)  are  most  useful  for  cutting  gears 
of  large  size,  in  which  tools  acting  on  the  formed  tool  or  mold- 
ing-generating principle  would  be  subject  to  too  heavy  cuts. 
Since  helical  gearing  is  generally  confined  to  small  and  medium- 
sized  work,  these  processes  are  unnecessary,  being  by  nature 
rather  slow  in  action,  and  dependent  for  their  accuracy  :on  the 
preservation  of  the  shape  of  easily  injured  points  of  compara- 
tively small  cutting  tools.  As  in  the  case  of  spur  gears,  the 
molding-generating  method  is  of  comparatively  recent  intro- 
duction, and  is  confined  almost  wholly  to  the  production  of 
teeth  of  involute  form. 

Machines  Using  Formed  Tools  in  a  Shaping  or  Planing  Oper- 
ation. —  With  the  twisted  teeth  in  gears  of  the  class  under  dis- 
cussion, it  is  evidently  necessary,  in  employing  shaping  or 
planing  operations,  to  give  a  rotary  movement  to  the  blank 
being  operated  on,  at  the  same  time  as,  and  in  the  proper  ratio 
with,  the  cutting  stroke  of  the  tool.  This  is  necessary  to  com- 
pel the  tool  to  follow  the  helix  on  which  the  teeth  of  the  gear  or 
the  worm  are  to  be  formed.  Attachments  have,  for  example, 

98 


METHODS  OF  CUTTING  TEETH 


99 


been  made  for  the  shaper,  giving  the  work  the  proper  motion  for 
cutting  helical  teeth. 

In  one  of  these  attachments  the  work  is  mounted  between 
centers  on  a  supplementary  bed,  fastened  to  the  work  table  of 
the  shaper.  The  faceplate  by  which  the  work  is  driven  from 
the  headstock  spindle  is  connected  to  that  spindle  by  an  indexing 
mechanism,  consisting  of  a  notched  plate,  with  a  locking  bolt  for 
holding  the  work  in  the  different  positions  for  the  different  num- 
bers of  teeth  required.  The  headstock  spindle  is  connected, 
by  spiral  gearing  and  a  set  of  change  gears,  with  a  pinion  oper- 
ated by  a  rack,  which  rack  is  fastened  to  the  shaper  ram.  It 
will  be  seen  that  this  connection  with  the  shaper  ram  will  give 
a  rocking  movement  to  the  headstock  spindle  and  the  work,  in 
unison  with  the  stroke  of  the  tool.  By  selecting  suitable  change 
gears,  this  rocking  movement  may  be  made  of  any  desired 
amplitude  for  a  given  length  of  stroke,  so  that  any  lead  of  helix 
or  spiral  desired  may  be  obtained.  Provision  is  made,  in  the 
mechanism  by  which  the  rack  is  attached  to  the  ram,  for  raising 
or  lowering  the  work  table  to  the  position  required  for  different 
diameters  of  work.  The  tool  is,  of  course,  fed  downward  by 
hand,  and  the  indexing  is  done  by  hand  also. 

In  another  shaper  attachment  a  spur  gear  keyed  to  the  head- 
stock  spindle  meshes  with  a  vertical  rack,  sliding  in  a  guide 
which  is  cast  integrally  with  the  headstock.  This  vertical  rack 
is  pivoted  to  a  block  which  slides  in  a  guide  attached  to  a  swiveling 
head,  so  that  the  guide  may  be  adjusted  to  any  angle.  This 
swiveling  head,  in  turn,  is  attached  to  a  bar,  which  is  fastened 
to  the  ram,  and  is  guided  on  ways  supported  by  a  framework 
at  the  back  of  the  headstock.  It  will  thus  be  seen  that  the 
forward  and  backward  movement  of  the  ram  will  impart  an  up- 
and-down  movement  to  the  rack,  which  will,  in  turn,  give  a 
rocking  movement  to  the  spindle  of  the  headstock,  and  the  work 
which  it  drives.  The  amplitude  of  this  rocking  can  be  increased 
or  diminished  by  setting  the  swiveling  guide  at  a  greater  or  less 
angle,  so  that  the  helices  of  various  leads  can  be  obtained. 
This  makes  the  use  of  change  gears  unnecessary.  The  indexing 
device  is  similar  in  principle  in  the  two  arrangements. 


100 


SPIRAL  GEARING 


It  will  seem  strange  at  first  thought,  perhaps,  to  describe 
the  cutting  of  worms  in  a  lathe  as  an  example  of  the  use  of 
formed  tools  in  shaping  or  planing  operations,  but  the  operation 
is  essentially  the  same  as  that  involved  in  shaping  spiral  gear 
teeth  by  means  of  the  attachments  described.  In  Fig.  i  imagine 
that  the  lead-screw  shown  is  of  such  steep  pitch  that  it  can  be 
rotated  by  pushing  the  carriage  backward  and  forward.  Under 
these  circumstances,  if  provision  is  made  for  reciprocating  the 
carriage  (corresponding  to  the  ram  for  the  shaper),  the  lead- 
screw  will  be  rotated  in  unison  with  it,  and  this  movement  will 
be  transmitted  through  change  gears  A,  B,  C  and  D  to  the  head- 


6-TOOTH  GEAR  OR  WORM  BEING  CUT 


GEARS  A,  B,  C  AND  D  ARE  CHOSEN 
TO  GIVE  THE  LEAD  DESIRED  FOR  THE 
SPIRAL  GEAR  OR  WORM 


DETAIL  OF  FACE- 
PLATE, SLOTTED 

FOR  INDEXING  A 
8 -TOOTH  SPIRAL 

GEAR  OR  WORM 

Machinery 


Fig.  i.    The  Lathe  Method  of  Planing  Helical  Teeth  in  Gears  or  Worms 

stock  spindle,  giving  a  rocking  movement  to  the  work.  The 
only  difference  in  the  two  cases  is  that  in  the  lathe  a  screw  of 
very  steep  pitch  would  be  used  to  change  the  reciprocating 
motion  of  the  tool  to  the  rocking  motion  required  by  the  work, 
while  in  the  case  of  the  shaper  the  more  natural  rack  and  pinion 
movement  is  employed. 

In  the  case  of  the  lathe,  of  course,  the  power  is  not  applied 
to  the  carriage  but  to  the  spindle.  For  that  reason  it  is  best 
adapted  for  cutting  spiral  gears  of  comparatively  small  lead,  or 
"worms"  as  they  are  ordinarily  called.  If  it  were  attempted  to 
cut  45-degree  spirals,  for  instance,  the  lead-screw  would  have  to 
be  speeded  up  so  fast,  as  compared  with  the  movement  of  the 
spindle,  that  the  driving  belt  would  be  unable  to  operate  the 
machine.  Special  lathes  have  been  built  for  cutting  steep 
worm  threads,  in  which  the  power  has  been  applied  to  the  lead- 


METHODS  OF  CUTTING  "TEFlfe  'J  IOI 

screw,  the  spindle  being  driven  from  it  through  the  change  gears. 
A  lathe  so  arranged  would  have  as  much  difficulty  in  cutting 
fine  pitches  as  the  ordinary  lathe  does  in  cutting  coarse  ones. 

Different  methods  of  indexing  may  be  used  for  the  lathe.  It 
will  be  noticed  that  in  Fig.  i  the  faceplate  used  has  the  same 
number  of  slots  as  the  required  number  of  teeth.  After  one 
tooth  space  has  been  cut,  the  work  can  be  removed,  and  re- 
placed again  between  the  centers  with  the  tail  of  the  dog  in 
another  slot.  After  this  space  has  been  completed,  the  next 
one  is  cut,  and  so  on  until  the  whole  six  are  finished.  Other 
methods  are  in  use,  such  as  slipping  of  change  gears  A  and  B 
past  each  other  a  certain  number  of  teeth,  as  determined  by 
calculation. 

Special  lathes  are  built  for  threading,  some  of  which  are  auto- 
matic in  their  action.  One  of  these  is  built  by  the  Automatic 
Machine  Co.,  Bridgeport,  Conn.  The  size  especially  adapted  to 
cutting  worms  is  provided  with  mechanism  for  duplicating  the 
action  of  a  manually-operated  lathe  engaged  in  threading.  After 
a  piece  of  work  has  been  placed  between  the  centers  and  the 
machine  has  been  started,  the  work  revolves,  and  the  carriage 
feeds  forward  until  the  proper  length  thread  has  been  cut;  then 
the  tool  is  withdrawn,  and  the  carriage  returns  to  begin  again 
on  a  new  cut  —  and  so  on  without  attention  from  the  operator. 
The  tool  is  fed  in  a  certain  suitable  amount,  at  the  beginning 
of  each  cut,  the  amount  of  this  feed  being  automatically  dimin- 
ished to  give  a  fine  finish  for  the  final  cuts.  When  the  depth  for 
which  the  tool  has  been  set  is  reached,  the  operation  of  the 
mechanism  is  automatically  arrested.  In  cutting  multiple- 
threaded  worms  in  this  machine,  multiple  tools  may  be  used, 
thus  avoiding  the  necessity  for  indexing  the  work.  As  many 
as  eight  cutting  tools  have  been  used  at  once  on  this  machine, 
giving  a  total  length  of  cutting  edge  of  8  inches. 

Machines  Using  Formed  Milling  Cutters.  —  The  formed  tool 
or  cutter  method  of  shaping  the  teeth  of  gears  is  generally  con- 
sidered as  being  one  in  which  the  tool  accurately  reproduces  its 
shape  in  the  tooth  space  it  forms.  This  is  true  in  cutting  straight- 
tooth  spur  gears,  and  in  planing  the  teeth  of  spiral  gears  by  the 


102 


SPIRAL  BEARING 


process  just  described.  It  is  not  exactly  true,  however,  of  any 
possible  process  of  milling  spiral  teeth.  This  is  best  seen  in 
Fig.  2.  In  the  three  cases  here  shown  we  have  first,  a  planer 
tool;  second,  a  disk  milling  cutter;  and  third,  an  end  milling 
cutter  —  all  formed  to  the  same  identical  outline,  and  cutting 
helical  grooves  of  the  same  lead  and  depth  in  blanks  of  the  same 
diameter.  The  section  in  each  case  is  a  plane  one,  taken  normal 
to  the  helix  at  the  pitch  line.  Of  course,  the  true  section  to  take 
would  be  that  of  the  helicoid  normal  to  the  helicoid  of  the  groove 
being  cut.  The  plane  in  which  we  have  taken  the  section,  how- 


ALL  SECTIONS  TAKEN  ON  LINE  X-X 

7 

/  \      FORM 


Fig.  2.     Comparison  of  the  Accuracy  of  Form  Reproduction  obtainable 
by  Formed  Planing  Tool,  Formed  Disk  Cutter  and  Formed  End-mill 

ever,  so  nearly  approximates  this  helicoid  that  the  error  is  negli- 
gible. 

The  planer  tool  necessarily  cuts  a  groove  of  the  same  shape 
as  its  outline,  the  plane  of  its  outline  being  the  same  as  the  plane 
of  the  section  shown.  The  disk  milling  cutter,  however,  inter- 
feres with  the  sides  of  the  groove  it  cuts.  This  interference  takes 
place  on  one  side  as  the  teeth  are  entering,  and  on  the  other  as 
the  teeth  are  leaving.  This  results  in  a  generating  action, 
which  takes  place  in  addition  to  the  simple  forming  action,  so 
that  the  tooth  cut  is  not  an  exact  duplicate  of  the  outline  of  the 
cutter.  In  the  case  of  the  formed  end-mill  there  is  also  an  inter- 


METHODS  OF  CUTTING  TEETH  103 

ference  of  the  same  kind  as  with  the  formed  disk  cutter,  but  it 
is  so  slight  as  to  be  absolutely  undetectable,  in  all  ordinary  cases. 
Its  presence  is  only  known  from  theoretical  considerations. 

In  spite  of  its  imperfect  reproduction  of  the  desired  form,  the 
disk  cutter  is  the  type  generally  used  for  milling,  since  it  may 
be  so  relieved  as  to  retain  its  shape  even  after  repeated  grinding. 
The  end-mill  type  of  formed  cutter  cannot  remove  so  much 
stock  in  a  given  time,  and  it  is  difficult  to  make  it  so  that  it  can 
be  ground  without  changing  its  form.  The  only  way  in  which 
this  grinding  can  be  practically  performed  is  by  the  use  of  some 
form  of  templet  grinding  machine.  The  formed  end-mill  is 
nevertheless  used  to  a  limited  extent. 

The  simplest  way  of  using  the  milling  process  for  cutting  hel- 
ical gears  or  worms  makes  use  of  the  universal  milling  machine. 
With  this  machine  the  work  and  the  feed-screw  of  the  table 
on  which  it  is  mounted  are  so  connected  by  means  of  gearing 
that  the  forward  feeding  gives  a  rotary  movement  to  the  work, 
producing  a  helix  of  the  required  lead.  The  mechanism  is  iden- 
tical in  principle  with  that  shown  in  Fig.  i  for  the  lathe,  the  only 
difference  being  that  in  the  milling  process  the  longitudinal 
movement  is  a  steady  feeding  motion,  made  once  for  each  tooth 
space,  instead  of  being  a  continuously  reciprocating  motion. 
The  simple  indexing  device  shown  in  Fig.  i  is  replaced  by  the 
more  elaborate  index  plate  and  worm-wheel  device  of  the  spiral 
head. 

This  mechanism  is  exemplified  in  the  Brown  &  Sharpe  univer- 
sal milling  machine  with  its  spiral  head.  The  work  has  to  be 
swung  at  an  angle  with  the  cutter  to  agree  with  the  helix  angle 
at  the  pitch  line.  This  is  done  by  swiveling  the  table  of  the 
universal  milling  machine  to  bring  the  work  to  the  proper  angle 
with  the  cutter.  In  most  makes  of  machines  it  is  inconvenient, 
if  not  impossible,  to  swivel  the  table  to  a  greater  angle  than 
45  degrees.  For  greater  angles  special  attachments  are  pro- 
vided for  swiveling  the  cutter,  leaving  the  table  in  its  normal 
position  at  right  angles  to  the  spindle  of  the  machine.  These 
various  attachments  allow  the  milling  machine  to  work  through- 
out a  wide  range  of  angles  for  helical  gears  and  worms,  the  only 


IO4 


SPIRAL  GEARING 


limitation  being  one  similar  to  that  imposed  on  worm  cutting 
in  the  lathe,  though  the  limitation  is  reversed.  For  worms  or 
gears  of  too  small  lead  as  compared  with  their  diameter,  the 
rotary  movement  of  the  blank  is  so  great  that  the  comparatively 
slow-moving  feed-screw  is  unable  to  speed  up  the  spiral  head 
mechanism  to  get  the  required  movement,  and  still  furnish  power 
enough  for  feeding  the  work  against  the  cutter. 

Points  Relating  to  the  Milling  of  Spiral  Gears.  —  Before 
describing  other  methods  for  cutting  the  teeth  in  spiral  or  helical 
gears,  we  shall  briefly  cover  the  essential  points  to  be  considered 
in  milling  these  gears  in  a  universal  milling  machine.  The  first 
point  to  be  considered  is  the  pitch  of  cutter  to  be  used.  The 
thickness  of  the  cutter  at  the  pitch  line  for  milling  spiral  gears 

should  equal  one-half  the  normal  cir- 
cular pitch  n  (see  Fig.  3).  If  a  cutter 
were  used  having  a  thickness,  at  the 
pitch  line,  equal  to  one-half  the  cir- 
cular pitch  P,  as  for  spur  gearing,  the 
spaces  between  the  teeth  would  be 
cut  too  wide,  thus  producing  thin 
teeth.  The  normal  pitch  varies  with 
the  angle  a  of  the  spiral;  hence,  the 
spiral  angle  must  be  considered  when 


Fig.  3.    Relation  between  Normal  selecting  a  Cutter.     The  CUtter  should 
and  Regular  Circular  Pitch  . 

be  of  the  same  pitch  as  the  normal 

diametral  pitch  of  the  gear  and  this  normal  pitch  is  found  by 
dividing  the  "real"  diametral  pitch  by  the  cosine  of  the  spiral 
angle.  To  illustrate,  if  the  pitch  diameter  of  a  spiral  gear  is  6.718 
and  there  are  38  teeth  having  a  spiral  angle  of  45  degrees,  the 
"real"  diametral  pitch  equals  38  -f-  6.718  =  5.656;  then,  the  nor- 
mal diametral  pitch  equals  5.656  divided  by  the  cosine  of  45  de- 
grees or  5.656  -5-  0.707  =  8.  A  cutter,  then,  of  8  diametral  pitch 
is  the  one  to  use  for  this  particular  gear.  This  same  result  could 
also  be  obtained  as  follows:  If  the  circular  pitch  P  is  0.5554,  the 
normal  circular  pitch  n  can  be  found  by  multiplying  the  circular 
pitch  P  by  the  cosine  of  the  spiral  angle.  For  example,  0.5554  X 
0.707  =  0.3927.  The  normal  diametral  pitch  is  then  found  by 


METHODS  OF   CUTTING  TEETH 


dividing  3.1416  by  the  normal  circular  pitch.     Thus  3'141     =  8, 

0.3927 

which  is  the  diametral  pitch  of  the  cutter. 

According  to  the  Brown  &  Sharpe  system  of  cutters  for  spur 
gears  having  involute  teeth,  eight  different  shapes  of  cutters 
(marked  by  numbers)  are  used  for  various  numbers  of  teeth  in 
gears  of  any  one  pitch.  When  the  diametral -pitch  is  known, 

ANGLE  OF  TEETH  WITH  AXIS  OF  GEAR 
5      10    15    20     25    30    35    40    45    50    55    60    65    70     75     80     85    90 


§30 
<35 

s: 

w  40 


45 


P* 


FORMULA: 

T-      N 
cos9  oc 

T=  number  of  teeth 
for  which  to  select 
cutter. 

N=number  of  teeth  in 

spiral  gear. 
««--angle  of  teeth  with 

axis  of  gear. 


Fig.  4.     Diagram  for  finding  Cutter  for  Milling  Spiral  Gears 

the  number  of  cutter  for  that  particular  pitch  must,  therefore, 
be  determined  as  explained  in  Chapters  I  and  II.  A  diagram 
and  table,  useful  in  this  connection,  are  given  in  the  following. 
Diagram  for  Finding  Cutter  for  Milling  Spiral  Gears.  —  A 
diagram,  Fig.  4,  has  been  prepared,  giving  directly  the  number  of 
cutter  to  be  used  for  a  given  number  of  teeth  and  a  given  spiral 
angle.  The  heavy  lines  drawn  in  the  diagram  are  division  lines 
between  the  fields  to  which  each  cutter  applies.  For  example, 


106  SPIRAL  GEARING 

suppose  the  angle  of  the  teeth  of  a  gear  is  37  degrees  with  its 
axis,  and  the  number  of  teeth  is  48.  The  point  A,  at  which  the 
horizontal  line  representing  the  number  of  teeth  and  the  vertical 
line  representing  the  angle  intersect,  falls  within  the  area  marked 
cutter  No.  2 ;  therefore,  a  No.  2  cutter  is  required  for  cutting  a 
48- tooth  spiral  gear  having  a  spiral  angle  of  37  degrees. 

Table  for  Selecting  Cutter  for  Milling  Spiral  Gears.  —  The 
" Table  for  Selecting  Cutter  for  Milling  Spiral  Gears"  gives  the 

value  of  the  factor  K  =  — —  which  enters  in  the  formula  for 

cos3  a 

finding  the  number  of  teeth  for  which  to  select  the  cutter  for 
miUing  spiral  gears.  The  table  is  used  as  follows:  Multiply  the 
actual  number  of  teeth  in  the  spiral  gear  to  be  cut  by  the  factor 
K,  as  given  in  the  table  opposite  the  angle  of  spiral.  The  product 
gives  the  number  of  teeth  for  which  to  select  the  cutter. 

Example.  — Angle  of  spiral  =  30  degrees;  number  of  teeth  in 
spiral  gear  =  18. 

Factor  K  for  30  degrees,  as  found  from  the  table,  equals  1.540. 
Then,  number  of  teeth  for  which  to  select  the  cutter  =  18  X 
1.540  =  28,  approximately.  Hence,  use  spur  gear  cutter  for 
28  teeth,  or  cutter  No.  4. 

Angular  Position  of  Table  when  Milling  Spiral  Gears.  —  In 
cutting  a  spiral  gear  in  a  milling  machine  as  ordinarily  arranged, 
it  is  necessary  to  set  the  table  to  the  helix  angle  in  order  that 
the  sides  of  the  cutter  may  not  interfere,  or  drag  in  the  cut;  but 
the  helix  angle  varies  with  the  depth,  being  greatest  at  the  top 
of  the  tooth,  less  at  the  pitch  line  and  still  less  at  the  bottom 
of  the  cut.  In  fact,  if  the  cut  were  deep  enough  to  reach  all  the 
way  to  the  center  of  the  piece  being  operated  on,  the  helix  angle 
would  become  zero,  or  parallel  to  the  center  line.  If  mechanics 
in  general  were  asked  what  would  be  the  proper  angle  at  which 
to  set  the  table,  they  would,  in  most  cases,  say  that  the  helix 
angle  at  the  pitch  line  would  be  the  one  to  determine  the  setting. 
This  setting  has  the  effect  of  making  the  width  of  the  cut  ex- 
actly right  at  the  pitch  line,  but  it  does  so  at  the  expense  of 
undercutting  and  weakening  the  teeth. 

It  has,  therefore,  been  frequently  stated  that  the  most  suit- 


METHODS  OF  CUTTING  TEETH 
Table  for  Selecting  Cutter  for  Milling  Spiral  Gears 


Angle  of 

j£ 

Angle  of 

|£ 

Angle  of 

/£ 

Angle  of 

X 

Spiral,  a 

Spiral,  a 

Spiral,  a 

Spiral,  a 

0°     o' 

I  .OOO 

21°      0' 

1.228 

42°     o' 

2.436 

63°     o' 

10.69 

0°3o' 

I  .OOO 

21°   30' 

I.24I 

42°  30' 

2-495 

63°  30' 

11.27 

1°      0' 

I  .OOI 

22°      Q' 

1.254 

43°     o' 

2-557 

64°     o' 

11.87 

i°3°' 

I  .OOI 

22°    30' 

1.268 

43°  3o' 

2.621 

-  64°  30' 

12.55 

2°      0' 

1  .002 

23°     o' 

1.282 

44°     o' 

2.687 

65°    o' 

13-25 

2°    30' 

1.003 

23°    30' 

1.297 

44°  30' 

2.756 

65°  30' 

14.03 

3°     o' 

1  .004 

24°     o' 

I.3I2 

45°     o' 

2.828 

66°    o' 

14.86 

3°  30' 

1.005 

24°    30' 

1.328 

45°  30' 

2.902 

66°  30' 

15.80 

4°     o' 

1.007 

25°    o' 

1-344 

46°     o' 

2.983 

67°    o' 

16.76 

4°  30' 

1  .009 

25°    30' 

1.360 

46°  30' 

3.066 

67°  30' 

17-85 

5°     o' 

I  .Oil 

26°     o' 

1-377 

47°     o'. 

3-I52 

68°    o' 

18.98 

5°  30' 

1.013 

26°  30' 

1-395 

47°  30'' 

3.242 

68°  30' 

20.33 

6°     o' 

1  .016 

27°     o' 

1.414 

48°     o' 

3.336 

69°    o' 

21.72 

6°  30' 

1.019 

27°  30' 

1-434 

48°  30' 

3-436 

69°  30' 

23-33 

7°     o' 

1.022 

28°     o' 

1-454 

49°     o' 

3-540 

70°    o' 

25.00 

7°  30' 

I  .026 

28°  30' 

1-474 

49°  30' 

3-650 

70°  30' 

26.97 

8°     o' 

1.030 

29°     o' 

1-495 

50°     o' 

3-767 

71°    o' 

28.97 

8°  30' 

1-034 

29°    30' 

I-5J7 

50°  30' 

3.887 

71°  30' 

31-40 

9°     o' 

1.038 

30°     o' 

1-540 

5i°    o' 

4.012 

72°    o' 

33-88 

9°  30' 

1.042 

30°  30' 

1-563 

5i°  3o' 

4.144 

72°  30' 

36.92 

10°     o' 

1.047 

3i°     o' 

1-588 

52°    o' 

4.284 

73°    o' 

40.00 

10°  30' 

1.052 

31°  30' 

1.613 

52°  30' 

4-433 

73°  30' 

43-88 

11°      0' 

1-057 

32°     o' 

i  .640 

53°    o' 

4-586 

74°    o' 

47-79 

n°  30'. 

1.062 

32°  30' 

1.667 

53°  3o' 

4.752 

74°  30' 

54-72 

12°      0' 

1.068 

33°     o' 

1.695 

54°    o' 

4.925 

75°    o' 

57-68 

12°    30' 

1.074 

33°  30' 

1.724 

54°  30' 

5.101 

75°  30' 

64-15 

13°     o' 

I.OSO 

34°     o' 

1-755 

55°    o' 

5-295 

76°    o' 

70-65 

13°  30' 

1.087 

34°  30' 

1.787 

55°  30' 

5-497 

76°  30' 

79.20 

14°     o' 

1.094 

35°     o' 

1.819 

56°    o' 

5-710 

77°    o' 

87.78 

14°  30' 

I  .  102 

35°  30' 

1-853 

56°  30' 

5-940 

77°  30' 

99-50 

15°     o' 

I.  110 

36°     o' 

1.889 

57°    o' 

6.190 

78°    o' 

in-  3 

15°  30' 

I.II8 

36°  30' 

1.926 

57°  30' 

6-435 

79°    o' 

144.0 

16°     o' 

I  .127 

37°     o' 

1.963 

58°    o' 

6.720 

80°    o' 

191.2 

16°  30' 

I.I36 

37°  30' 

2.003 

58°  30' 

7.010 

81°    o' 

261  .4 

17°     o' 

I  .145 

38°     o' 

2.044 

59°    o' 

7.321 

82°    o' 

370.6 

17°  30' 

I-I54 

38°  30' 

2.086 

59°  30' 

7.650 

83°    o' 

552-1 

18°    o' 

I.I63 

39°     o' 

2.130 

60°    o' 

8.000 

84°    o' 

876.4 

18°  30' 

I.I72 

39°  30' 

2.176 

60°  30' 

8.380 

85°     o' 

1509.0 

19°    o' 

I,l82 

40°     o' 

2.225 

61°    o' 

8.780 

86°    o' 

2940  .  o 

19°  30' 

I-I93 

40°  30' 

2.275 

61°  30' 

9.209 

87°    o' 

6990  .  o 

20°      O' 

1  .  2O<: 

41°    o' 

2  .  326 

62°    o' 

9.6^8 

20°    W 

I  .  2l6 

41°  30' 

**   o 
2.380 

62°  30' 

V  -  **3 

10.  160 

'      ow 

*TA          O 

•  •  o^ 

io8 


SPIRAL  GEARING 


able  angle  (and  the  one  most  likely  to  produce  the  best  results) 
at  which  to  set  the  table  of  the  milling  machine  when  milling 
spiral  gears  is  that  corresponding  either  to  the  diameter  of  the 
gear  measured  at  the  bottom  of  the  space,  or  to  the  diameter 
measured  at  the  working  depth.  The  reason  invariably  ad- 
duced for  this  is,  as  just  mentioned,  that,  if  the  angle  chosen  is 
the  angle  of  the  spiral  measured  on  the  pitch  cylinder  of  the 
gear,  an  undue  amount  of  undercutting,  and  therefore  weaken- 
ing, of  the  teeth  will  occur,  owing  to  an  excessive  amount  of 
interference  with  the  sides  of  the  teeth  on  the  part  of  the  cutter; 


:  ANGLE  OF  TABLE 


Machinery 


Fig.  5.     Shapes  of  Teeth  Obtained  by  Setting  the  Table  at  Different 
Angles,  the  Cutter  and  the  Lead  remaining  the  Same 

and  that,  therefore,  a  somewhat  smaller  angle  should  be  selected 
to  reduce  these  effects. 

To  determine  whether  there  was,  practically,  anything  in  this 
idea  or  not,  some  experiments  were  recently  made  on  a  spiral 
gear,  the  immediate  object  of  the  experiments  being  to  find  out 
what  the  effect  of  altering  the  angle  of  setting  of  the  milling 
machine  table  was  upon  the  shape  of  the  tooth  cut. 

The  experiments  were  made  upon  a  cast-iron  gear,  with  a 
pitch  diameter  of  4.242  inches,  and  designed  for  24  teeth,  the 
diametral  pitch  (corresponding  to  the  normal  circular  pitch) 
being  8.  The  correct  cutter  to  use  was  determined  by  the  form- 
TV 
ula  Ne  =  — — ,  this  cutter  being  No.  3  in  each  of  the  cases  dealt 


METHODS  OF  CUTTING  TEETH 


IOQ 


with.  The  experiments  consisted  of  cutting  six  teeth  in  the 
gear  blank,  all  being  of  the  same  depth,  the  angle  of  setting  of 
the  table  of  the  milling  machine  being  different  in  each  of  the 
six  cases.  The  spiral  angle  measured  on  the  pitch  cylinder  was 
45  degrees,  the  lead  of  the  spiral  being  13.32  inches,  for  which 
the  gears  of  the  spiral  dividing-head  were  arranged.  The  six 
spirals  chosen  were  at  angles  of  45,  44,  43,  42,  41  and  40  degrees, 
each  tooth  being  formed  by  two  cuts  at  one  angle,  the  lead  of 
the  spiral  remaining  the  same  throughout  the  series  of  tests. 
It  should  be  here  noted  that  43  degrees  is  the  angle  which  corre- 
sponds to  the  diameter  measured  at  the  bottom  of  the  space. 

The  profiles  of  the  teeth  taken  as  sections  normal  to  the  spiral 
on  the  pitch  surface  are  indicated  in  Fig.  5,  the  profiles  being 
Table  of  Observed  Tooth  Dimensions 


Depth  at  which  Tooth  is  Measured 

Table 
Setting, 

o 

0.050 

O.IOO 

0.125 

0.150 

0.200 

0.250 

Degrees 

Width  of  Tooth 

45 

0.104 

0.145 

0.185 

O.2OO 

0.203 

O.  221 

0.239 

44 

O.IO2 

0.144 

0.181 

0.195 

O.2O2 

O.22O 

0.238 

43 

0.099 

0.142 

0.176 

0.188 

O.2OO 

0.218 

0.236 

42 

0.094 

0.135 

0.168 

O.lSo 

0.196 

0.215 

0.234 

4i 

0.087 

0.128 

0.158 

O.I7I 

0.185 

O.2II 

0.232 

40 

0.078 

0.115 

0.146 

0.158 

O.I7I 

0.205 

0.230 

drawn  accurately  to  scale  —  three  times  full  size.  The  various 
widths  of  the  teeth  at  different  depths  were  obtained  as  accur- 
ately as  possible  by  means  of  a  Brown  &  Sharpe  gear-tooth  vernier 
caliper.  These  widths  are  given  in  the  accompanying  table. 
Of  course,  it  will  be  readily  seen  that  although  great  care  was 
exercised  in  securing  measurements  that  would  be  as  accurate  as 
possible,  the  dimensions  given  above  may  be  incorrect  by  about 
one  or  two  thousandths  inch,  but  not  more. 

In  regard  to  the  shapes  of  the  teeth,  it  will  be  noticed  that  the 
45-degree  tooth  is  slightly  undercut  at  the  root,  while  the  other 
teeth  do  not  show  any  undercutting  whatever.  The  under- 
cutting referred  to  in  the  45-degree  tooth  amounts  to  a  reduction 


HO  SPIRAL  GEARING 

in  width  below  the  widest  part  of  the  tooth  of  about  o.oio 
inch. 

The  deductions  drawn  from  the  results  of  these  tests  are: 

1.  That  the  practice  of  setting  the  table  at  an  angle  less  than 
the  spiral  or  helix  angle  measured  on  the  pitch  surface  is  justified; 
although  this  angle  should  not  be  less  than  the  spiral  or  helix 
angle  measured  at  the  bottom  of  the  tooth. 

2.  That  a  cutter  for  a  larger  number  of  teeth  than  that  given 

N 

by  the  formula  Ne  =  — —   could  probably  be  employed,  in 
cos3  a 

order  to  counteract  the  flattening  and  widening  effect  of  the  cutter 
with  an  angle  as  indicated  above. 

In  spite  of  the  good  reasons  given  for  setting  the  table  to  the 
angle  determined  by  the  root  of  the  teeth,  it  is  common  practice 
to  set  the  table  to  the  spiral  angle  of  the  teeth  at  the  pitch  line. 
In  any  case,  the  angle  is  determined  by  first  obtaining  the  tan- 
gent of  the  angle,  and  then  finding  the  corresponding  angle  from 
a  table  of  tangents.  For  example,  if  the  pitch  diameter  of  the 
gear  is  4.46  and  the  lead\)f  the  spiral,  20  inches,  the  tangent 

equals  — &-J. —  =  0.700,  which  is  the  tangent  of  35  degrees; 

20 

therefore  the  table  should  be  swiveled  35  degrees  from  its  position 
at  right  angles  to  the  spindle. 

Milling  the  Spiral  Teeth.  —  After  a  tooth  space  has  been 
milled,  the  cutter  should  be  prevented  from  dragging  through  it 
when  being  returned  for  another  cut.  This  can  be  done  by 
lowering  the  blank  slightly,  or  by  stopping  the  machine  and  turn- 
ing the  cutter  to  such  a  position  that  the  teeth  will  not  touch 
the  work.  If  the  gear  has  teeth  coarser  than  10  or  12  diametral 
pitch,  it  is  well  to  take  a  roughing  and  a  finishing  cut.  When 
pressing  a  spiral  gear  blank  on  the  arbor,  it  should  be  remembered 
that  it  is  more  likely  to  slip  when  being  milled  than  a  spur  gear, 
because  the  pressure  of  the  cut,  being  at  an  angle,  tends  to  rotate 
the  blank  on  the  arbor. 

Specialized  Forms  of  Milling  Machines  for  Cutting  Spirals  by 
the  Formed  Cutter  Method.  —  The  principle  of  the  universal 
milling  machine  for  cutting  spiral  gears  and  worms  has  been  ap- 


METHODS  OF   CUTTING  TEETH  ill 

plied  to  the  design  of  various  special  machines  for  the  same 
purpose.  The  specialization  of  these  machines  includes  making 
the  spiral  and  indexing  mechanisms  integral  parts  of  the  tool,  so 
that  they  have  a  much  greater  capacity  for  taking  heavy  cuts 
than  is  the  case  where  they  are  merely  attachments. 

In  Fig.  6  is  shown  a  diagram  of  the  index  worm  connections 
of  a  universal  gear  cutting  machine  made  by  J.  E.  Reinecker,  of 
Chemnitz- Gablenz,  Germany,  as  arranged  for  cutting  helical  gears 
by  the  formed  milling  process.  The  machine  is  arranged  on  the 
general  lines  of  a  milling  machine,  except  that  the  work  spindle 
is  at  the  top  of  the  column,  and  the  cutter  spindle  on  the  knee. 

The  cutter  is  driven  by  an  internal  gear  of  large  diameter  and 
is  mounted  on  a  swivel  table  which  can  be  set  to  the  required  helix 
angle.  The  form  of  cutter  slide  will  provide  for  any  angle  up 
to  30  degrees.  For  greater  angles  it  is  replaced  with  a  slide  which 
can  be  rotated  to  any  angle  throughout  the  whole  circle. 

The  screw  which  feeds  the  cutter  slide  along  the  knee  is  driven 
from  cone  pulley  D,  through  a  vertical  shaft  and  gear  connec- 
tions. Cone  pulley  D  is  also  connected  with  change  gearing  F, 
which  is,  in  turn,  connected  with  the  index  worm,  so  as  to  rotate 
index  wheel  G  and  the  work,  for  any  desired  helix.  The  prin- 
ciple of  this  is  the  same  as  in  the  universal  milling  machine, 
change  gears  F  acting  the  same  as  the  change  gears  used  to  con- 
nect the  spiral  head  with  the  table  feed-screw.  Now  the  worm- 
wheel  G  is  used  for  indexing,  as  well  as  for  rotating  the  work  for 
the  helix,  in  unison  with  the  feeding  of  the  cutter  slide.  The 
way  in  which  these  two  motions  are  imparted  to  G  without  inter- 
fering with  each  other,  may  be  understood  from  the  following 
description. 

At  H  are  mounted  the  change  gears  by  which  the  indexing  is 
accomplished.  These  gears  drive  bevel  gear  /.  Index  worm  K, 
meshing  with  index  worm-wheel  G,  is  mounted  on  a  hoflow 
sleeve,  keyed  fast  to  the  bevel  gear  L.  Shaft  M  carries  a  hub 
with  projecting  pivots  on  its  right-hand  end,  on  which  are 
mounted  bevel  pinions  N.  Shaft  M  is  driven  by  worm-wheel  0, 
connected  with  the  feed  of  the  slide  cutter  through  change  gears 
F.  Gears  /,  L  and  N  form  a  differential  mechanism  of  the  well- 


112 


SPIRAL  GEARING 


Wl 

J 


METHODS  OF  CUTTING  TEETH  113 

known  " jack-in-the-box"  type.  The  action  of  this  mechanism 
is  such  that  if  shaft  M  be  at  rest,  change  gears  at  H  may  be 
operated  for  the  indexing,  transmitting  the  motion  from  gears  / 
to  L  through  pinions  N  as  idlers,  thus  revolving  index  worm  K. 
On  the  other  hand,  with  the  indexing  mechanism  still  and  the 
cutter  slide  feeding,  the  movement  thus  imparted  to  shaft  M 
may  be  transmitted  (by  the  rolling  of  pinions,^  on  stationary 
bevel  gear  /,  and  the  consequent  rotation  of  bevel  gear  Z),  to 
worm  Kj  and  thence  to  worm-wheel  G  and  the  work.  It  will 
thus  be  seen  that  the  indexing,  and  the  rotation  for  the  helical 
cutting,  can  take  place  independently  of  each  other.  But  more 
than  this,  the  two  motions  can  be  operated  together  without 
interference.  In  fact,  either  of  the  motions  imparted  to  shaft  M 
or  gears  at  H  may  be  stopped  or  reversed  independently,  and  each 
will  have  its  proper  influence  on  the  index  wheel  and  the  work. 

With  this  understanding  of  the  differential  mechanism,  the 
operation  of  the  machine  is  easily  comprehended.  Change  gears 
H  are  connected  through  a  one-revolution  friction  trip  with  the 
main  driving  shaft.  The  cutter,  set  at  the  proper  angle,  is  fed 
forward  through  the  work,  which  is  rotated  by  change  gears  F, 
shaft  M  and  worm  K,  at  the  proper  rate  to  cut  the  proper  helix. 
The  cutter  is  then  dropped  down  to  clear  the  work  (provision  for 
this  being  made  in  the  machine) ,  and  returned,  ready  to  begin  on 
a  new  tooth.  The  indexing  mechanism  is  then  tripped  by  hand, 
and  the  work  is  rotated  into  position  for  the  new  tooth  by  change 
gears  at  H,  gear  /  and  worm-wheel  K.  This  is  repeated  until 
the  gear  is  done. 

In  the  machine  described  power  is  applied  to  the  feed-screw, 
from  which  the  work  is  rotated  through  change  gearing.  This 
arrangement  is  best  for  helices  of  great  lead.  When  it  comes  to 
milling  helical  gears  with  small  leads,  or  worms,  it  is  necessary 
to  use  the  lathe  principle  and  apply  the  power  to  rotating  the 
work,  the  longitudinal  feed  being  driven  from  the  work  spindle 
through  change  gearing. 

Specialized  Form  Milling  Machines  for  Herringbone  Gears.  — 
A  machine  for  helical  gear  cutting,  but  provided  with  some 
special  features,  is  used  by  C.  E.  Wuest  &  Co.,  Seebach,  Zurich, 


114  SPIRAL  GEARING 

Switzerland,  for  cutting  herringbone  gears  of  a  special  form,  in 
which  it  is  unnecessary  to  cut  the  two  halves  separately,  in  separ- 
ate sections,  as  is  the  usual  case.  (See  preceding  chapter.)  The 
cuts  are  staggered  so  that  the  teeth  on  one  side  run  into  the 
spaces  on  the  other,  in  such  a  way  as  to  permit  cutting  them  with 
rotary  cutters  without  having  one  side  interfere  with  the  other. 
The  machine  for  doing  this  is  built  on  a  very  simple  plan.  It 
consists  of  a  vertical  spindle  carrying  the  work,  which  is  indexed 
by  power.  The  indexing  wheel  is  connected  by  the  usual  change 
gearing  with  the  two  vertical  slides  on  which  the  cutters  are 
mounted  on  either  side.  These  cutters  work  simultaneously, 
one  feeding  downward  to  cut  the  upper  half,  while  the  other  is 
feeding  upward  to  cut  the  lower  half. 

There  is  another  specialized  form  of  herringbone  gear  made 
by  Andre  Citroen  &  Co.,  Paris,  France.  The  teeth  of  these 
gears  are  shaped  by  an  end  milling  cutter,  guided  by  suitable 
mechanism  to  produce  the  continuous  "wavy"  form  of  herring- 
bone teeth  characteristic  of  these  gears  (see  Fig.  i,  Chapter  III). 
This  process  also  has  the  advantage  of  not  requiring  the  blank 
to  be  made  in  two  pieces.  The  same  principle  has  been  applied 
by  the  builders  to  the  cutting  of  herringbone  bevel  gears. 

Other  manufacturers  make  use  of  the  formed  end-mill  to  a 
limited  extent.  The  arrangement  devised  by  Gould  &  Eberhardt 
for  milling  large  helical  gears  in  the  lathe  used  this  form  of 
cutter,  and  the  worms  or  spiral  gears  which  drive  the  racks  of 
the  Sellers  drive  planers,  made  by  at  least  one  of  our  prominent 
planer  builders,  are  cut  by  end-mills  in  a  specialized  milling 
machine  of  simple  design,  made  especially  for  this  purpose. 

Automatic  Machines  for  Milling  with  Formed  Cutters.  —  A 
number  of  full  automatic  machines  have  been  built  in  an  exper- 
imental way  for  milling  spiral  gears  with  formed  cutters.  They 
have  usually  been  modelled  after  the  automatic  spur  gear  cutter. 
Evidently  the  mechanism  has  to  be  considerably  more  com- 
plicated. The  first  complication  involved  is  due  to  the  fact  that 
the  index  wheel  must  be  under  the  influence  of  both  the  helical 
and  the  indexing  movements,  as  in  the  Reinecker  machine  in 
Fig.  6.  The  differential  gearing  there  shown  is  the  arrangement 


METHODS  OF  CUTTING  TEETH 


generally  used  to  effect  the  combination  of  these  movements  in 
the  automatic  helical  gear  cutter. 

Another  complication  is  introduced  by  the  necessity  for  re- 
lieving the  cutter  on  its  return  stroke,  after  finishing  the  forward 
feed  through  the  blank.  Backlash  in  the  rotating  mechanism 
between  the  cutter  slide  and  the  work  so  alters  the  position  of 
the  cutter  and  the  work,  on  the  return  stroke, -that  the  latter 
will  drag  on  the  one  side  of  the  groove  it  has  just  cut,  unless  it 


HELICAL  PATH  OF\  f , 
CUTTING  TOOTH  /  V 


Machinery 


Fig.  7.     The  Molding-generating  Principle  arranged  to  Employ  a  Cutter 
having  a  Helical  Shaping  Action  Cutting  Teeth  in  a  Solid  Blank 

is  separated  slightly  from  it.  This  has  been  done  in  various 
ways  in  the  various  machines  built;  in  some  cases  by  mounting 
the  cutter  on  a  supplementary  holder  which  rocks  back  out  of 
the  way  on  the  return  stroke,  and  in  other  cases  by  withdraw- 
ing the  work  by  mechanism  provided  for  the  purpose. 

These  various  complications  seem  to  have  militated  against 
the  commercial  success  of  the  automatic  spiral  gear  cutting 
machine  to  such  an  extent  that,  so  far  as  the  author  knows,  but 
one  of  the  various  designs  built  has  ever  left  the  shop  where  it 
was  made. 


n6 


SPIRAL  GEARING 


Molding-generating  Principle :  Planing  Operations.  —  Pass- 
ing by  the  templet,  odontographic  and  describing-generating 
principles,  for  the  reasons  mentioned  in  the  introduction  to  this 
chapter,  we  come  to  the  molding-generating  principle.  This  is 
applied  to  helical  gears  in  the  same  way  as  to  spur  gears,  with 
such  modifications  as  are  necessary  to  allow  for  the  helical  shape 
of  the  teeth.  In  Fig.  7  the  forming  cutter  and  the  blank  to  be 
cut  are  rolled  together,  while  the  forming  cutter  is  reciprocated 


ALTERNATIVE 
FORMING  RACK 


Machinery 


Fig.  8.  A  Rack  with  Teeth  set  on  an 
Angle  Operating  by  Impression  on 
the  Molding-generating  Principle 
to  Form  Teeth  in  a  Helical  Gear 


Fig.  9.  Shaper  Tools  Representing 
Teeth  of  an  Imaginary  Rack  Oper- 
ating on  the  Molding-generating 
Principle  in  a  Helical  Gear 


axially.  In  combination  with  the  axial  movement,  however,  the 
cutter  has  to  be  given  a  rocking  movement  about  its  center  line,  so 
that  its  teeth  will  follow  the  path  of  the  dotted  lines  shown,  which 
indicate  the  helix  of  the  spiral  gear  which  the  cutter  represents. 

In  Fig.  8  the  forming  rack  has  teeth  set  on  the  same  angle  as 
the  helix  angle  desired  in  the  gear  being  formed.  The  rolling 
of  a  plastic  blank  over  this  forming  rack  will  form  in  the  blank 
helical  teeth  of  the  shape  desired.  A  top  view  of  the  rack  is 


METHODS  OF  CUTTING  TEETH  117 

shown,  which  will  make  this  clearer.  Instead  of  the  forming 
rack  shown  by  the  full  lines,  one  like  that  shown  in  the  dotted 
lines  may  be  used,  whose  teeth  coincide  with  those  of  the  first, 
but  which  moves  in  a  direction  at  right  angles  to  the  direction  of 
its  teeth.  If  this  dotted  rack  is  moved  at  such  a  rate  of  speed 
that  its  teeth  always  coincide  with  those  of  the  rack  shown  in 
full  lines,  they  will  evidently  both  form  teeth  of  exactly  the 
same  shape  in  the  blank. 

In  Fig.  9  we  have  the  dotted  rack  of  the  top  view  of  Fig.  8, 
shown  engaged  in  the  operation  of  generating  the  teeth  of  a  gear 
identical  with  that  in  Fig.  8.  This  view  has  been  taken  at  an 
angle  so  as  to  show  the  normal  view  of  the  rack.  If  the  proper 
relative  rates  of  rotation  of  the  work  and  movement  of  the  rack 
are  maintained  in  Figs.  8  and  9,  and  the  normal  sections  of  the 
racks  in  each  case  are  the  same,  the  gears  generated  will  be  the 
same.  It  is  evident  in  Fig.  9  that  the  teeth  of  the  rack  may  be 
replaced  by  shaper  or  planer  tools  7\  and  T2,  which  may  be  used 
in  forming  teeth  on  the  blank  by  rotating  the  gear  and  moving 
the  tools  endwise,  in  the  proper  ratio  prescribed  by  the  condi- 
tions in  Fig.  8. 

Fig.  9  is  interesting  in  that  it  hints  at  the  principle  on  which 
the  action  of  the  helical  gearing  is  based.  As  drawn,  it  shows 
very  plainly  the  action  of  the  well-known  Sellers  drive  for  planers. 
It  will  be  noted  that  for  a  short  space  the  rack  teeth  exactly  fill 
the  outline  of  the  gear  tooth.  Contact  between  the  gear  and  the 
rack  takes  place  on  straight  lines  running  diagonally  across  the 
plane  faces  of  the  rack  teeth. 

/  Practical  application  has  been  made  of  the  principle  shown  in 
Fig.  9.  The  Bilgram  spiral  gear  planing  machine  involves  this 
principle.  The  work  is  mounted  on  a  spindle  carried  in  a  head, 
which  swivels  about  a  vertical  axis  so  that  it  may  be  set  to  the 
helix  angle  of  the  gear  being  cut.  The  cutting  tool,  having  a 
shape  to  represent  a  tooth  of  the  imaginary  generating  rack,  is 
carried  by  a  ram  which  works  in  and  out,  cutting  on  the  return 
stroke.  This  ram  is  carried  by  a  head  which  is  fed  along  the  bed 
of  the  machine.  This  longitudinal  feeding  of  the  ram-carrying 
head  is  connected  with  the  rotary  movement  of  the  work  spindle 


Il8  SPIRAL   GEARING 

by  change  gearing,  in  the  proper  ratio  for  the  case  in  hand,  so 
that  the  gear  will  roll  with  the  movement  of  the  head  just  as 
it  would  if  it  were  acting  under  the  influence  of  the  imaginary 
rack,  one  of  whose  teeth  is  represented  by  the  cutting  tool.  The 
conditions  are  thus  exactly  the  same  as  in  Fig.  9. 

Under  these  conditions,  if  the  machine  is  set  properly,  the  cut- 
ting tool  will  start  to  work  at  one  side  of  the  blank,  and  pass 
through  it,  feeding  at  the  end  of  each  successive  stroke,  with 
the  work  rolling  in  such  a  way  as  to  form  a  tooth  space  of  the 
proper  shape.  This  action  is  modified  somewhat  by  the  method 
of  indexing  adopted.  The  arrangement  used  indexes  the  work 
at  every  stroke,  so  that  when  the  tool  has  once  passed  through 
the  work,  the  gear  is  entirely  completed,  every  tooth  having 
been  worked  on.  This  indexing  movement  and  the  rolling 
motion  required  for  the  generating  are  superimposed  on  each 
other  by  suitable  mechanism  so  that  neither  interferes  with  the 
other. 

It  may  be  mentioned  incidentally  that  this  machine  is  the 
only  one  known  to  the  author  in  which  all  the  requirements  for 
theoretical  accuracy  in  cutting  helical  gears  have  been  taken 
care  of.  There  is  a  minute,  although  actual,  error  involved  in 
even  the  otherwise  perfect  hobbing  process  for  cutting  these  gears. 

The  Hobbing  Modification  of  the  Molding-generating  Princi- 
ple. —  Instead  of  using  the  shaper  or  planer  tool  to  take  the  place 
of  the  teeth  of  the  imaginary  rack  shown  in  Fig.  9,  a  hob  may 
be  used  in  the  same  way  as  for  hobbing  spur  gears.  This  con- 
dition is  shown  in  Fig.  10,  which  should  be  compared  with 
Fig.  9.  The  upper  or  plan  view  best  shows  the  respective  angu- 
lar settings  of  the  work  and  the  hob.  The  hob  is  set  at  an  angle 
with  the  line  of  movement  of  the  imaginary  rack  equal  to  its 
own  helix  angle,  as  for  spur  gears.  The  gear  being  cut  is  set  at 
an  angle  with  this  same  line  equal  to  its  own  helix  angle,  so  that 
in  this  case  (in  which  both  gear  and  hob  are  right-hand)  they 
are  set  at  an  angle  to  each  other  equal  to  the  difference  between 
the  helix  angles.  If  the  hob  represented  by  the  worm  in  the 
diagram  is  revolved  in  the  direction  shown,  its  teeth  will  have 
the  same  outline  and  the  same  movement  as  the  teeth  of  an 


METHODS  OF  CUTTING  TEETH 


imaginary  rack,  moving  in  the  direction  shown.  If  the  work  be 
revolved  in  the  proper  ratio  with  the  hob,  the  latter  will  form 
the  teeth  in  the  former  in  the  same  way  that  the  imaginary  rack 
would,  provided  it  is  fed  progressively  through  the  work  in  the 
direction  of  line  XX. 

This  necessity  for  feeding  the  hob  through  the  work  introduces 
an  added  complexity  to  the  machine  in  the  case -of  spiral  gears, 
beyond  that  needed  for  the  spur  gear  hobbing  machine.  To 
understand  this,  suppose  that  in  Fig.  10  the  spindle  mechanism 


GEAR  BEING  CUT 


WORM  REPRESENTING  THE  HOB 
WHICH  IS  CUTTING  THE  GEAR 


GEAR  BEING  CUT 


lOF  IMAGINARY  RACK  V 

I  HELIX  ANGLE  Of  H 

\  \DIRECTION  OF  FEEDING 
X  MOVEMENT  OF  HOB 

MAGINARY  RACK  WHICH  FORMS 

THE   GEAR    BY  THE    MOLDING 

GENERATING  PROCESS.      ITS  TEETH 

COINCIDE  WITH  THOSE  OF  THE 

HOB,  WHEN  THE  LATTER  IS  SET 

AS  SHOWN. 

Machinery 


Fig.  10.    Molding-generating  Method  for  Cutting  Spiral  Gears  as 
Exemplified  in  the  Hobbing  Process 

is  stopped,  so  that  both  the  spindle  and  work  cease  to  revolve. 
To  make  it  possible  to  feed  the  hob  through  the  work  in  the 
direction  of  line  XX  without  having  the, teeth  of  the  one  strike 
against  the  other,  it  will  be  necessary  to  revolve  either  the  work 
or  the  hob.  Suppose  that  the  work  be  connected  by  change 
gearing  with  the  feed-screw  of  the  cutter  slide,  so  that  it  is  re- 
volved as  the  cutter  is  fed  up  or  down.  Under  these  conditions 
the  cutter  may  be  moved  through  the  work  freely,  the  latter 
revolving  to  allow  the  cutter  to  pass.  Not  only  must  the  work 
revolve  in  a  definite  relation  with  the  feeding  of  the  cutter  slide, 


120 


SPIRAL  GEARING 


but  the  work  must  also  revolve  in  unison  with  the  cutter  or  hob, 
as  for  spur  gears.  It  must  then  be  so  connected  with  the  cutter 
and  with  the  cutter-slide  feed-screw  that  it  will  be  under  the 
influence  of  either  or  both  of  them,  without  any  interference  of 
the  two  movements  with  each  other.  This  connection  is  usually 
made  by  a  " jack-in-the-box"  or  differential  mechanism,  exactly 
identical  in  principle  with  that  shown  in  Fig.  6  for  combining 
the  indexing  and  helical  feeding  movements  for  revolving  the 
work  in  the  Reinecker  universal  machine.  In  the  case  of  the 


CUTTER  SLIDE  FEED  SCREW 


CUTTER  DRIVING  SHAFT 


DRIVING  SHAFT  OF  MACHINE 


DIFFERENTIAL,  OR  JACK-IN-THE-BOX  GEARING 

WORM  FOR  REVOLVING  WORK, TABLE 


HANGE  GEARS  FOR  LEAD  OF  SPIRAL 


HANGE  GEARS  FOR  RATE  OF  FEED 


Machinery 


Fig.  ii. 


Typical  Arrangement  of  Gearing  for  Spiral  Gear  Hobbing 
Machine 


spiral  gear  bobbing  machine  we  have  a  helical  feeding  move- 
ment and  a  cutter  spindle  movement  to  combine  for  revolving 
the  work. 

A  typical  arrangement  of  the  mechanism  used  for  this  purpose 
is  shown  in  diagrammatic  form  in  Fig.  n.  Power  is  applied  to 
the  machine  through  driving  shaft  G.  The  bevel  gears  shown 
connect  this  driving  shaft  with  vertical  shaft  //,  by  means  of 
which  the  hob  is  driven.  Change  gears  shown  connect  shaft 
G  with  shaft  E.  Considering  for  the  time  being  that  worm- 


METHODS  OF  CUTTING  TEETH  121 

wheel  B  and  the  attached  bevel  gear  D  are  stationary,  the  rota- 
tion of  E  and  the  cross-arm  A  keyed  to  it  will  cause  bevel  gears 
C  to  roll  around  on  stationary  gear  D,  thereby  revolving  gear  L 
and  shaft  F  to  which  it  is  keyed,  thus  rotating  the  work  table. 
The  change  gears  connecting  G  and  E  are  selected  to  give  the 
proper  ratio  of  movement  between  the  hob  or  cutter  spindle, 
and  the  work  table,  to  agree  with  the  number  of  threads  in  the 
hob  and  the  number  of  teeth  in  the  gear  being  cut.  The  cutter- 
slide  feed-screw  K  is  connected  by  change  gears  with  shaft  7, 
which  is,  in  turn,  connected  through  the  clutch  and  the  bevel 
gears  shown  with  shaft  E.  The  clutch  furnishes  the  means  of 
stopping  and  starting  the  feed,  and  the  change  gears  serve  to 
give  the  rate  of  feed  desired.  Change  gears  are  also  provided 
connecting  bevel  gear  M  on  feed-screw  K,  with  worm  L,  which 
drives  worm-wheel  £,  running  loosely  on  shaft  E.  By  this 
means,  supposing  for  the  moment  that  shaft  E  and  its  attached 
cross-arm  A  are  stationary,  the  rotation  of  the  feed-screw  is 
communicated  through  the  change  gears  to  worm-wheel  B  and 
its  attached  bevel  gear  D,  which,  driving  bevel  pinions  C  on  their 
stationary  studs,  revolve  gear  L,  and  with  it  shaft  F  and  the 
worm  driving  the  work  table.  In  this  way,  by  selecting  suitable 
change  gears,  the  work  may  be  revolved  to  agree  with  the  length 
of  the  lead  of  the  spiral  on  which  its  teeth  are  formed,  so  that 
the  cutter  may  be  fed  up  and  down  through  it  without  inter- 
fering with  the  teeth. 

It  will  be  seen  that  the  mechanism  shown  in  Fig.  n  may  be 
arranged  to  connect  the  hob  and  the  work  in  the  proper  ratio, 
as  for  cutting  spur  gears,  and  also  for  connecting  the  feed-screw 
and  the  work  in  the  proper  ratio  as  for  cutting  spiral  gears  in 
the  milling  machine.  This  mechanism  not  only  performs  these 
two  functions  separately,  but  it  will  perform  them  together,  as 
well,  so  that  either  the  feed  or  the  cutter  revolving  mechanism 
may  be  started,  stopped  or  reversed  independently  of  the  other 
movement,  and  the  work  will  still  be  properly  controlled  under 
all  conditions.  The  mechanism  shown  is  not  that  invariably 
used,  but  it  is  typical  of  the  arrangement  employed  in  many 
hobbing  machines  designed  for  cutting  helical  gearing. 


122  SPIRAL  GEARING 

Field  of  the  Robbing  Process  for  Helical  Gears.  —  There  are 
some  limitations  to  the  hobbing  process  for  cutting  helical  gears. 
It  is  not  particularly  successful  in  the  cutting  of  gears  of  such 
small  lead  and  great  helix  angle  that  they  would  be  classed  as 
worms,  rather  than  spiral  gears.  For  such  cases  the  rate  of 
rotation  which  has  to  be  given  the  blank  is  so  great  in  propor- 
tion to  the  downward  feed  of  the  cutter  by  which  the  rotation  is 
effected  (through  the  change  and  differential  gearing)  that  it  is 
almost  impossible  to  drive  it,  the  difficulty  being  the  same  in 
kind,  though  reversed  in  direction,  as  that  met  with  in  cutting 
very  steep  pitches  in  the  lathe.  By  a  slight  complication  of  the 
machine,  however,  mechanism  could  be  introduced  to  overcome 
this  difficulty,  and  make  the  hobbing  machine  universal  for  all 
kinds  of  gears  within  its  range. 

In  discussing  the  hobbing  processes  for  cutting  spur  gears,  it 
is  often  stated  that  its  field  is  not  yet  definitely  determined.  It 
may  be  said,  on  the  whole,  that  there  is  no  such  indefiniteness 
in  regard  to  the  field  of  the  hobbing  machine  for  cutting  helical 
gears.  With  a  well-constructed  machine  and  with  hobs  of 
proper  shape,  spiral  gears  can  be  cut  more  accurately  and  cheaply 
by  this  method  than  by  any  other  process  known.  There  are 
none  of  the  mechanical  difficulties  of  indexing  and  relieving  to 
be  taken  care  of  as  is  the  case  in  automatic  machines  working 
on  the  formed  cutter  process;  and  there  are  none  of  the  uncer- 
tainties as  to  tooth  shape  due  to  interference  met  with  in  cutting 
a  helical  groove  with  a  formed  cutter,  as  shown  in  Fig.  2.  There 
has  been  some  little  difficulty  in  getting  the  correct  shape  of 
teeth  by  the  hobbing  process,  due  to  the  elasticity  of  the  mechan- 
ism connecting  the  hob  and  the  work,  and  to  errors  in  the  con- 
struction of  the  hob  itself.  These  difficulties,  however,  will 
surely  disappear  with  further  experience  and  investigation. 

Apparently  the  recent  rapid  development  of  the  hobbing 
process  for  cutting  spiral  gears  is  the  solution  of  a  problem  which 
has  long  seemed  somewhat  perplexing.  The  flexibility  of  the 
spiral  gear,  and  the  numerous  advantages  of  the  herringbone  or 
the  twisted-tooth  spur  gear  for  transmitting  great  power  noise- 
lessly and  smoothly  even  at  high  velocities,  have  long  been  ap- 


METHODS  OF  CUTTING  TEETH  123 

predated,  but  their  extended  use  has  waited  for  the  development 
of  some  accurate  and  inexpensive  method  of  forming  helical  teeth. 

Calculating  Gears  for  Generating  Spirals  on  Robbing  Ma- 
chines. —  From  time  to  time  formulas  have  been  developed  for 
calculating  the  gears  to  be  used  for  generating  spiral  gears. 
Those  published  in  the  past,  however,  have  applied  only  to  cer- 
tain types  of  gear-hobbing  machines.  In  the  following  a  formula 
is  given  which  was  first  published  in  MACHINERY,  December, 
1911,  and  which  is  applicable  to  any  type  of  gear-hobbing  ma- 
chine, and  which  is  simpler  to  use  than  any  formula  that  had 
been  published  up  to  that  time.  In  developing  this  formula, 
simple  arithmetical  expressions  have  been  made  use  of,  as  far  as 
possible,  in  order  to  make  it  especially  useful  to  the  practical  man. 

In  order  to  clearly  understand  the  use  of  any  formula,  it  is 
necessary  to  know  something  of  the  principles  involved.  Fig.  1 2 
shows  a  top  view  of  a  standard  bobbing  machine  (the  No.  3 
Farwell)  designed  for  cutting  spur  gears.  Before  dealing  with 
the  change-gear  ratios  for  spiral  work,  it  will  be  well  to  have  the 
methods  for  cutting  spur  gears  properly  understood.  Assume 
the  hob  to  be  single  threaded.  It  is  evident  that  for  each  rev- 
olution of  the  hob,  the  gear  being  cut  must  move  one  tooth. 
Therefore,  the  hob  revolves,  for  each  revolution  of  the  blank, 
as  many  times  as  there  are  teeth  to  be  cut.  To  cut  44  teeth, 
the  table  must  be  geared  to  revolve  once  for  every  44  revolutions 
of  the  hob. 

The  bevel  gearing  at  Z),  Fig.  12,  has  a  ratio  of  3  to  i,  the  worm 
at  E  is  double-threaded,  and  the  worm-wheel  F  has  40  teeth. 
Hence,  the  shaft  B  must  revolve  3  X  44  times  for  each  revolution 
of  the  table,  and  the  worm  shaft  C  must  revolve  20  times  for  each 
revolution  of  the  table.  Hence,  we  have: 

Revolutions  of  B  _  3  X  44 
Revolutions  of  C         20 

Inverting  this  ratio  to  get  the  change-gear  ratio  required  to 
obtain  this  result,  we  have: 

20       _  Product  of  No.  of  teeth  in  driving  gears 
3  X  44      Product  of  No.  of  teeth  in  driven  gears 


124 


SPIRAL  GEARING 


8UV30  30NVHO 


METHODS  OF  CUTTING  TEETH 


I2S 


In  the  following  formulas  we  will  designate  the  product  of 
the  number  of  teeth  in  the  driving  gears  Pt  and  the  product  of 
the  number  of  teeth  in  the  driven  gears  p. 

Should  we  use  a  double- threaded  or  triple- threaded  hob,  the 
gear  we  are  cutting  must  revolve  two  or  three  teeth  for  each  rev- 
olution of  the  hob;  in  other  words,  the  speed  of  the  table  is  in- 
creased directly  as  the  number  of  threads  on  the  hob,  so  we 
must  multiply  the  number  of  teeth  in  the  driving  gears  by  the 
number  of  threads  on  the  hob,  giving  us  this  formula: 

20  X  No.  of  threads  on  hob  _  P 
3  X  No.  of  teeth  to  be  cut       p 

A  similar  formula  may  be  worked  out  in  this  way  for  any  type 
of  gear  hobber. 

Generating  Spirals.  —  For  each  revolution  of  the  table  the 
head  carrying  the  hob  feeds  down  a  certain  distance  across  the 
face  of  the  blank,  this  distance  varying  from  o.oio  to  0.150  inch 
in  common  practice.  To  fully  understand  the  following  dis- 
cussion, the  action  of  the  machine,  as  illustrated  in  Figs.  13  to  16, 
inclusive,  should  be  noted.  In  Fig.  13  is  shown  the  generation 
of  a  right-hand  spiral  gear  with  a  right-hand  hob;  in  Fig.  14 
a' left-hand  spiral  gear  with  a  right-hand  hob;  in  Fig.  15  a  left- 
hand  spiral  gear  with  a  left-hand  hob;  and  in  Fig.  16  a  right- 
hand  spiral  gear  with  a  left-hand  hob.  In  each  of  these  illus- 
trations the  direction  of  rotation  of  the  table  is  indicated  by  the 
arrow  showing  the  direction  of  rotation  of  the  gear  being  cut. 
The  direction  of  rotation  of  the  hob  is  also  indicated  by  an  arrow 
showing  the  direction  of  rotation  of  its  shaft.  In  Figs.  13  and 
15,  where  a  gear  is  cut  with  a  hob  of  the  same  "hand/'  the  angle 
a,  as  indicated,  equals  the  difference  between  the  tooth  angle 
and  the  thread  angle  of  the  hob.  In  Figs.  14  and  16,  where  the 
gear  and  the  hob  are  of  different  "hand,"  the  angle  a  equals 
the  sum  of  the  tooth  angle  and  the  thread  angle  of  the  hob. 
After  this  preliminary  introduction,  we  are  ready  to  deal  intelli- 
gently with  the  problem  in  hand. 

Assume  the  spiral  gear  shown  in  Fig.  17  to  have  sixty-four 
teeth.  As  indicated,  the  gear  has  a  left-hand  spiral  and  we  will 


126 


SPIRAL  GEARING 


assume  that  it  is  cut  with  a  left-hand  hob.  A  single-threaded 
hob  cutting  a  spur  gear  would  revolve  sixty-four  times  for 
one  revolution  of  the  table;  but  since  in  this  case  the  teeth 
are  helical  and  the  hob  travels  downward  a  certain  distance, 
the  position  of  the  gear  tooth  must  be  advanced  somewhat  for 
every  revolution  with  relation  to  the  hob.  In  other  words, 


Machinery 


Fig.  13.     Cutting  a  Right-hand  Spiral  Gear  with  a  Right-hand  Hob 


Machinery 


Fig.  14.     Cutting  a  Left-hand  Spiral  Gear  with  a  Right-hand  Hob 

if  the  hob  revolves  sixty-four  times,  sixty-four  teeth  will  have 
passed  by,  but  the  blank  is  not  in  the  same  position  as  at 
the  beginning. 

In  Fig.  17  G  represents  the  position  of  the  hob  axis  at  the 
beginning  of  the  cut  and  H  the  position  of  the  hob  axis  after  the 
hob  has  made  sixty-four  revolutions.  This  shows  that  the  blank 
must  make  more  than  one  revolution  in  this  case.  If  we  were 
cutting  a  left-hand  spiral  gear  with  a  right-hand  hob,  as  shown 
in  Fig.  14,  the  blank  would  have  to  make  less  than  one  complete 


METHODS  OF  CUTTING  TEETH 


127 


revolution  for  each  sixty-four  revolutions  of  the  hob,  the  blank 
in  this  case  being  revolved  in  the  opposite  direction.  It  will 
thus  be  seen  that  when  cutting  a  gear  of  the  same  "hand"  as 
the  hob,  the  table  must  revolve  slightly  faster  than  it  would  have 
to  do  when  cutting  a  spur  gear  with  the  same  number  of  teeth; 
but  when  the  hob  and  the  gear  are  of  opposite  "hand, "  the  table 
must  revolve  more  slowly  than  when  cutting  a  spur  gear.  This 


Machinery 


Fig.  15.     Cutting  a  Left-hand  Spiral  Gear  with  a  Left-hand  Hob 


Machinery 


Fig.  16.    Cutting  a  Right-hand  Spiral  Gear  with  a  Left-hand  Hob 

has  an  important  bearing  upon  the  formula  we  are  about  to 
construct. 

To  gear  the  machine  properly  we  must  first  find  the  ratio 
according  to  which  the  table  is  required  to  lag  behind  or  lead 
ahead  of  its  natural  speed  relative  to  the  hob.  In  the  first 
formula  devised  by  the  author  for  the  bobbing  of  spiral  gears, 
the  ratio  was  arrived  at  by  considering  the  number  of  revolutions 
made  by  the  hob,  while  the  table  makes  one  full  revolution. 


128 


SPIRAL  GEARING 


The  formula  thus  constructed  for  the  No.  i  Farwell  gear-hobbing 
machine  is: 

30  X  No.  of  threads  on  hob P 

No.  of  teeth  ±  [  (feed  X  tan  of  angle)  -f-  circ.  pitch]  p 
This  applies  only  to  one  particular  machine.  A  later  formula 
designed  for  the  No.  3  Farwell  machine,  as  shown  in  Fig.  12, 
considers  the  number  of  table  revolutions  required  while  the  hob 
revolves  a  sufficient  number  of  times  to  represent  one  revolu- 
tion of  the  table,  if  we  were  cutting  a  spur  gear: 


20  ± 


2O 


Pitch  circumference  -f-  (feed  X  tan  of  angle) 
(3  X  No.  of  teeth)  -=-  No.  of  threads  on  hob 


P_ 
P 


Machinery 


Fig.  17.     Diagram  showing  Advance  Required  in  Table  Motion  when 
Cutting  a  Left-hand  Spiral  Gear  with  a  Left-hand  Hob 

Being  called  upon  to  derive  another  formula  to  be  used  for  the 
new  No.  3  Farwell  universal  hobbing  machine,  it  occurred  to  the 
originator  of  these  formulas,  that  a  formula  adapted  to  all 
hobbing  machines  would  avoid  much  confusion.  In  the  follow- 
ing is  given  the  process  by  which  such  a  formula  was  derived; 
the  result  is  a  simpler  formula  than  any  previously  used. 

Universal  Formula.  —  The  "  lead  "  of  a  spiral  gear  is  the 
axial  length  of  the  blank  in  which  one  spiral  tooth  makes  a  com- 
plete turn  around  the  blank.  Now,  in  hobbing  a  gear  with  a 
width  of  face  exactly  equal  to  the  lead,  it  is  evident  that  the 
blank  must  gain  or  lose  one  complete  revolution  as  compared 
with  the  number  of  revolutions  that  would  be  made  in  cutting 
a  spur  gear  with  the  same  width  of  face  and  using  the  same  feed 
per  revolution  of  the  blank.  Assume  that  it  is  desired  to  cut  a 


METHODS  OF   CUTTING  TEETH  129 

30-tooth,   io-pitch,  right-hand  spiral  gear  of  45-degree  angle, 
using  a  single-threaded  right-hand  hob  and  feeding  -£$  inch  across 
the  face  of  the  blank  for  each  revolution  of  the  blank. 
The  rule  for  finding  the  lead  of  a  spiral  gear  is: 

Pitch  circumference  X  cot  of  tooth  angle  =  lead. 

To  get  the  pitch  circumference,  first  find  the  pitch  diameter; 
the  rule  for  finding  this  in  a  spiral  gear  is: 

Pitch  diameter  of  spur  gear  -f-  cos  of  tooth  angle  =  pitch 
diameter  of  spiral  gear  with  the  same  number  of  teeth  and 
pitch. 

A  30-tooth,  io-pitch  spur  gear  would  have  a  pitch  diameter 
of  3  inches.  Referring  to  a  table  of  trigonometrical  functions 
it  will  be  found  that  the  cosine  of  45  degrees  is  0.70711;  then, 
3  -f-  0.70711  =  4.242  inches,  which  is  the  pitch  diameter  of  the 
spiral  gear.  Multiplying  this  by  3.1416  gives  13.3267  inches, 
which  is  the  pitch  circumference  of  the  spiral  gear.  Since  the 
cotangent  of  45  degrees  is  exactly  i,  multiplying  by  this  gives 
the  same  quantity  (13.3267  inches)  as  the  lead. 

The  next  step  is  to  find  how  many  times  the  blank  must  re- 
volve while  the  hob  feeds  13.3267  inches  across  its  face.  Since 
the  feed  is  ^  incn  (°-°3I25)  f°r  eacn  revolution,  we  can  divide 
by  0.03125  or  multiply  by  32  to  get  the  number  of  revolutions. 
This  gives  426.454  revolutions.  The  table  has  been  traveling 
faster  in  relation  to  the  hob  than  would  be  the  case  in  cutting  a 
spur  gear  with  the  same  number  of  teeth;  in  fact,  the  table  has 
gained  exactly  one  revolution  on  the  hob.  In  other  words,  the 
table  speed  in  cutting  this  spiral  gear  is  to  the  table  speed  in 
cutting  an  equivalent  spur  gear  as  426.454  is  to  425.454.  From 
this  we  may  construct  the  following  formula: 

Lead  -r-  feed        _  required  table  revolutions 
(Lead  -5-  feed)  —  i       normal  table  revolutions 

For  a  gear  of  opposite  "hand"  from  that  of  the  hob  the  sign 
would  be  changed  to  +  in  this  formula.  Use  the  —  sign  only 
when  gear  and  hob  are  of  the  same  "hand." 

By  adding  a  4 2 6- tooth  gear  to  the  drivers  and  a  4 2 5- tooth 
gear  to  the  driven  gears  in  the  regular  combination  used  to  cut 


130  SPIRAL  GEARING 

a  3<>tooth  spur  gear,  we  would  get  approximately  the  desired 
ratio,  but  for  greater  accuracy  we  can  carry  the  figures  to  a  few 
decimal  places  and  factor: 

42,645  _  8529  _  3  X  2843 
42,545      8509      67  X  127 

but  2843  is  a  prime  number.    We,  therefore,  try 

4265  __853 
4255      851 
but  853  is  a  prime  number.     We,  therefore,  try 

4264  =  2132  ^  4  X  533 
4254      2127      3  X  709 

but  709  is  a  prime  number.  Hence  we  must  make  another 
slight  change  and  try  again,  remembering  that  whatever  change 
is  made  in  the  numerator  must  be  exactly  duplicated  in  the 
denominator  to  maintain  the  ratio  as  nearly  as  possible.  The 
dropping  of  all  decimals  would  cause  a  very  small  error,  but 
dropping  them  from  one  side  only  would  cause  a  great  error. 
We  find  upon  trial  that 

426  _  2  X  3  X  71 

425  ~5  X5  X  17 

Multiplying  this  with  the  change-gear  combination  ordinarily 
used  to  cut  spur  gears  with  30  teeth,  we  have  the  gear  com- 
bination required  for  any  gear-hobbing  machine  used  for  cutting 
this  gear.  Thus,  on  the  No.  3  Farwell  universal  hobbing  ma- 
chine, the  spur-gear  ratio  for  cutting  30  teeth  is  f  -§-,  which  multi- 

2  X  "3  X  7i                  3  X  7i 
plied  by * '—  gives  • — ' and  arranging  this  ratio 

'  5X5  X  i7S        5X5X17 
in  convenient  gear  sizes,  we  have: 

24  X  71  _  product  of  teeth  of  driving  gears 
40  X  85      product  of  teeth  of  driven  gears 
It  will  be  noted  that  the  last  operation  before  factoring  was  to 
divide  by  the  feed.     Should  prime  numbers  be  encountered  re- 
peatedly in  trying  to  factor,  it  is  possible  to  get  altogether  new 
figures  to  work  with,  by  making  a  slight  change  in  the  feed  and 
dividing  into  the  lead  again. 


METHODS  OF  CUTTING  TEETH  131 

Having  found  the  gears,  set  the  feed  for  exactly  ^  inch  per 
revolution,  see  that  the  table  is  revolving  in  the  right  direction, 
and  tilt  the  hob  spindle  to  bring  the  thread  angle  to  45  degrees 
and  the  machine  is  ready  for  business. 

Recapitulation  and  General  Remarks.  —  The  general  formula 
for  gearing  any  hobbing  machine  for  generating  spiral  gears  is 
thus: 

L  +  F  P  _  5 

in  which 

L  =  lead  of  spiral; 

F  =  feed  per  revolution; 

P  =  product  of  driving  gears  for  cutting  spur  gears  with 

same  number  of  teeth; 
p  =  product  of  driven  gears  for  cutting  spur  gears  with 

same  number  of  teeth; 

S  —  product  of  driving  gears  for  cutting  spiral  gears; 
s  =  product  of  driven  gears  for  cutting  spiral  gears. 

Use  +  sign  when  gear  and -hob  are  of  opposite  "hand,"  and 
—  sign  when  they  are  of  the  same  "hand." 

In  cutting  teeth  at  large  angles  it  is  desirable  to  have  the  hob 
the  same  hand  as  the  gear,  so  that  the  direction  of  the  cut  will 
come  against  the  movement  of  the  blank,  but  at  ordinary  angles 
one  hob  will  cut  both  right-  and  left-hand  gears. 

The  actual  feed  of  the  cutter  depends  upon  the  angle  of  the 
teeth  as  well  as  on  the  vertical  movement  of  the  hob.  This 
is  obtained  by  dividing  the  vertical  feed  by  the  cosine  of  the 
tooth  angle;  thus: 

-^ — 5  =  0.043  inch  actual  feed. 
0.70711 

The  last  computation  need  not  be  made  except  to  see  that  we 
are  not  figuring  on  too  heavy  a  cut,  as  it  has  nothing  to  do  with 
the  gearing  of  the  hobbing  machine.  In  setting  up  a  hobbing 
machine  for  spiral  gears,  care  should  be  taken  to  see  that  the 
vertical  feed  does  not  trip  until  the  machine  has  been  stopped  or 
the  hob  has  fed  down  clear  of  the  finished  gear.  Should  the 


132  SPIRAL   GEARING 

feed  stop  while  the  hob  is  still  in  mesh  with  the  gear  and  revolv~ 
ing  at  the  ratio  required  to  generate  a  spiral,  the  hob  will  cut  into 
the  teeth  and  spoil  the  gear. 

Should  the  thread  angle  of  the  hob  be  exactly  equal  to  the 
tooth  angle  of  the  spiral  gear,  and  both  hob  and  gear  be  the 
same  "hand,"  the  axis  of  the  hob  spindle  will  be  at  right  angles 
to  the  axis  of  the  gear.  This  is  in  conformity  with  the  rule  that 
when  hob  and  gear  are  of  the  same  "hand,"  the  hob  spindle  is 
set  at  the  tooth  angle  minus  the  thread  angle  of  the  hob.  In 
cutting  a  spiral  gear  to  take  the  place  of  a  worm-wheel,  it  is  pos- 
sible to  use  the  same  hob  that  was  used  in  cutting  the  worm- 
wheel.  This  would  be  a  case  where  it  is  not  necessary  to  tilt 
the  hob  spindle.  Sometimes  multiple-threaded  hobs  are  used 
in  order  to  make  the  thread  angle  approximately  equal  to 
the  tooth  angle,  when  it  is  desired  to  cut  spiral  gears  with 
machines  on  which  the  hob  spindle  swivels  through  only  a 
small  angle. 

Examples  of  Calculations.  —  As  an  example  of  the  application 
of  the  formula  given  for  finding  the  gears  for  spiral  gear  hobbing, 
assume  that  two  spiral  gears  are  to  be  cut  on  a  gear-hobbing 
machine.  Gear  No.  i  has  30  teeth,  24. 549-inch  lead  and  a  feed 
of  gV  inch.  The  change  gears  used  on  the  machine  for  cutting 
a  spur  gear  with  30  teeth  have  48  (driving  gear)  and  60  (driven 
gear)  teeth,  respectively.  The  hob  and  gear  are  of  the  same 
"hand." 

Gear  No.  2  has  60  teeth,  49.o98-inch  lead  and  is  cut  with  a 
feed  of  ~IQ  inch.  The  change  gears  used  to  cut  a  spur  gear  with 
60  teeth,  on  this  machine,  have  48  and  40  teeth,  for  the  driving 
gears,  and  60  and  80  teeth,  for  the  driven  gears.  The  hob  and 
gear  are  of  the  same  "hand." 

In  the  problems  given  the  data  are  thus  as  follows : 

30-tooth  6o-tooth 

Gear  Gear 

L 24.549        L 49.098 

F W4              F Me 

P 48               P 40X48 

p 60                p 60X80 

The  same  notation  as  in  the  formula  just  given  is  used. 


METHODS  OF  CUTTING  TEETH  133 

Calculations  for  Thirty-tooth  Gear.  —  By  inserting  the  values 
given,  we  find  that: 

L-T-  F       _  589.176 
(L  +  F)—  i  ~  588.176 

The  ratio  written  above  can  be  simplified  to  the  form  ||f . 
Factoring,  we  have: 

589  _  19  X  31 
588      12  X  49 

Now,  multiply  this  value  with  the  ratio  of  the  gears  for  a  30- 
tooth  spur  gear: 

19  X  31  x  48  _  76X31 
12  X  49      60      60  X  49 

Having  obtained  the  gears  that  should  be  used,  we  may  now 
investigate  what  lead  these  gears  will  give.     Apparently  they 
will  not  give  the  exact  lead  desired,  as  we  have  used  an  approx- 
imate ratio  instead  of  the  exact  one. 
To  prove,  assume  F  =  •£%  and  solve  for  L. 
L  +  F 


(L  -s-  F)  -  i      588 

From  this  we  find  L  =  24.541,  which  is  very  nearly  equal  to 
the  required  lead. 

Calculations  for  Sixty-tooth  Gear.  —  By  proceeding  in  the 
same  way  for  the  60- tooth  gear  we  have : 

L  +  F       _  785.568 
(L-5-/0-I      784-568 
We  then  factor  the  fraction  ||f ,  thus: 

785  _  5  X  i57 
784      4  X  196 

As  157  is  a  prime  number,  and  gives  too  large  a  number  of 
teeth  for  any  of  the  gears  in  the  train,  we  try  \ ||-  which  ratio 
is  very  nearly  equivalent  to  that  required. 

784  _  49  X  16 
783      29  X  27 

Multiply  this  value  with  the  ratio  of  the  gears  for  a  6o-tooth 
spur  gear: 


134  SPIRAL  GEARING 

49  X  16       40  X  48  _  49  X  32 
29  X  27       60  X  80       29  X  135' 

Possibly  the  i35-tooth  gear  is  impracticable,  on  account  of 
being  too  large,  in  which  case  the  other  combination  must  be 
tried. 

If  the  lead  resulting  from  the  gears  found  is  calculated  in  the 
same  manner  as  in  the  previous  case,  we  find  that 

L  =  49.001. 

Influence  of  Small  Changes  in  the  Ratio  on  the  Lead.  —  It  is 

interesting  to  note  that  a  comparatively  slight  change  in  the 

r     .     -p 

ratio  — zir—    -   makes  a  very  decided  change  in  the  lead 

(L  -s-  f)  -  i 

obtained.  To  illustrate,  assume  that  in  the  first  example  given 
the  ratio  -f-ff  =  1.001701  were  changed  to  1.002;  let  us  see  what 
effect  this  change  would  have  on  the  lead  obtained  (F  =  £%) : 


(L  +  F)  -  i 


=  i. 002. 


If  we  solve  for  L  in  this  equation  we  find  L  =  20.875,  which  is 
a  very  different  lead  from  the  one  we  wish  to  obtain. 

Advantage  of  Differential  Mechanism  on  Gear-hobbing  Ma- 
chines in  Calculating  Change  Gears.  —  When  generating  helical 
gears  on  hobbing  machines  without  a  differential,  the  required 
ratio  which  combines  index  and  feed  gears  must  be  calculated 
to  a  great  many  decimals,  as  otherwise  a  large  error  will  result 
which  will  impair  the  accuracy  of  the  gears.  It  frequently  hap- 
pens that  the  required  ratio  consists  of  prime  numbers,  especially 
when  cutting  right-  and  left-hand  gears  with  one  hob.  To  pro- 
duce correct  helical  gears  with  their  axes  standing  parallel  to 
each  other,  the  errors  for  the  right-  and  left-hand  spirals  must 
be  absolutely  the  same,  otherwise  there  will  not  be  a  bearing  on 
the  whole  length  of  the  teeth.  In  fact,  exactly  the  same  condi- 
tions exist  with  helical  gears  as  with  spur  gears.  If,  for  instance, 
the  teeth  of  one  of  two  spur  gears  stood  at  an  angle  of  only  a  few 
seconds  with  its  axis,  the  bearing  would  be  at  one  end  of  the 
teeth  only. 


.METHODS  OF  CUTTING  TEETH  135 

Furthermore,  if  the  hobbing  machine  has  a  differential,  it  is 
not  necessary  to  have  a  right-  and  left-hand  hob  when  cutting 
any  angle  up  to  30  degrees;  on  the  contrary,  a  higher  efficiency 
is  obtained  when  using  only  one  hob  for  both  right-  and  left-hand 
spirals.  The  reason  for  this  is  very  simple;  if  there  is  any  dis- 
tortion in  hardening,  the  right-hand  hob  will  be  different  from 
the  left-hand. 

It  has  been  mentioned  before  that  the  ratio  must  be  calculated 
to  several  decimals  when  cutting  the  gears  on  machines  without 
a  differential.  The  belief  of  many  mechanics  that  the  ratios 
and  errors  obtained  by  formulas  are  alike  for  all  hobbing  ma- 
chines, with  or  without  differential  mechanism,  is  entirely  erro- 
neous. There  is  a  great  difference  between  the  two  ratios.  In 
the  one  case  the  ratio  represents  the  value  of  the  indexing  and 
the  helical  movement,  and  the  slightest  change  of  the  " driver," 
viz.,  numerator,  will  cause  a  great  error  if  the  "driven/'  viz., 
denominator,  is  not  also  changed  in  the  same  proportion.  In 
the  other  case,  i.e.,  with  the  differential,  the  ratio  obtained  refers 
to  the  angle  or  helical  movement  only,  and  adds  or  subtracts 
itself  automatically  to  or  from  the  ratio  of  the  indexing  gears. 
The  indexing  gears  required  for  cutting  helical  gears  are  given 
on  a  chart  and  can  be  read  off  the  same  as  for  spur  gears.  This 
is  impossible  without  the  differential.  The  difference  between 
the  two  ratios  is  explained  in  the  following  example. 

Example.  —  Gear,  48  teeth;  10  pitch;  20  degrees;  ^  inch 
feed  per  revolution  of  table. 

Gear  ratio  of  machine  with  differential  for  20  degrees  = 
1.2052784. 

If  we  deduct  i  from  the  third  decimal  which  is  5,  and  omit 
the  rest,  we  have  1.204  =  ratio  for  19  degrees  58  minutes  42  sec- 
onds; i.e.,  i  minute  18  seconds  difference. 

This  shows  how  slight  the  error  would  be  if  we  were  to  change 
the  third  decimal;  in  practice  the  change  is  made  on  the  fifth 
decimal,  and  the  error  almost  eliminated. 

For  the  same  pitch,  number  of  teeth,  angle  and  feed,  the 
gear  ratio  for  one  of  the  machines  without  a  differential  equals 
1.2517385.  If  here  we  were  to  deduct  i  from  the  third  decimal 


136  SPIRAL  GEARING 

and  omit  the  rest,  the  result  would  be  that  instead  of  generating 
teeth  the  material  would  simply  be  milled  off  from  the  blank. 
This  is  explained  as  follows:  Gear  ratio  for  20  degrees  is  1.2517385. 
When  deducting  i  from  the  third  decimal  we  obtain  1.250,  which 
is  the  spur-gear  ratio. 

The  Schuchardt  &  Schiitte  gear-hobbing  machines  are  pro- 
vided with  a  differential  which  on  the  new  type  of  machines  is 
independent  of  the  feed  and  indexing;  in  other  words,  when 
changing  the  number  of  teeth,  or  feed,  or  from  right-  to  left- 
hand  gear,  no  calculation  is  required.  Thus  the  great  advantage 
of  the  differential  mechanism  is  that  the  helical  movement  is 
not  disturbed  whatever  when  the  number  of  teeth  is  increased 
or  decreased  or  the  feed  is  changed.  Suppose  we  intend  to  gen- 
erate helical  gears  with  30,  40,  56  and  60  teeth,  of  which  those 
having  40  and  60  teeth  are  left-hand,  and  those  having  30  and 
56  teeth  are  right-hand;  the  spiral  angle  is  15  degrees;  the  pitch 
is  10.  The  material  is  supposed  to  be  cast  iron;  therefore  ^g- 
inch  feed  per  revolution  of  the  table  would  be  selected  as  the 
proper  one.  All  the  gears  are  to  be  cut  with  one  right-hand 
lo-pitch  hob.  In  calculating  the  change  gears  used  when  gen- 
erating these  gears  on  the  Schuchardt  &  Schiitte  machine  but  a 
few  minutes  will  be  required,  the  following  formula  being  used: 

Constant  X  sine  of  angle  X  pitch 

^ =  ratio. 

i 

0.3524  X  0.25882  X  io  _  912    _  19  X  48  _  driving  gears 
i  1000      20  X  50       driven  gears 

On  machines  not  provided  with  a  differential  mechanism, 
every  gear  of  the  same  pitch,  with  only  a  different  number  of 
teeth,  must  be  calculated  for  separately,  and  the  slightest  change 
in  the  feed  will  require  a  separate  calculation.  A  change  in  the 
formula  must  also  be  made,  if  right-  and  left-hand  gears  with  the 
same  number  of  teeth  are  cut  with  one  hob. 

The  differential  is  also  of  great  importance  when  cutting 
worm-gears  with  a  taper  hob.  Worm-gears  for  worms  with 
multiple  threads  ought  to  be  generated  with  taper  hobs  if  high 
efficiency  is  required. 


CHAPTER  V 
HOBS  FOR  SPUR  AND   SPIRAL  GEARS 

Robbing  vs.  Milling  of  Gears.  —  The  adverse  criticism  of  the 
gear-hobbing  process  has  been  the  cause  of  many  interesting  in- 
vestigations, and  one  of  the  most  important  of  these  has  been 
the  comparative  study  of  the  condition  of  the  surfaces  produced 
by  the  hob  and  by  the  rotary  cutter.  In  making  such  a  com- 
parative study,  it  is  necessary  that  the  investigator  possess  the 
required  practical  knowledge,  and  also  that  he  be  willing  to  admit 
a  point,  even  though  his  favorite  processes  may  suffer  by  the 
comparison. 

Feed  Marks  Produced  by  Rotating  Milling  Cutters.  —  While 
both  the  gear-hobbing  machine  and  the  automatic  gear  cutter 
use  rotating  cutting  tools,  the  operations  cannot  be  placed  on 
a  common  basis  and  considered  as  similar  milling  operations, 
although  they  may,  to  a  certain  extent,  be  compared  as  such. 
In  comparing  the  quality  of  the  surfaces  produced  by  the  two 
processes,  consider  first  the  milled  surface  produced  by  an  ordi- 
nary rotary  cutter.  This  surface  has  a  series  of  hills  and  hollows 
at  regular  intervals,  the  spacing  between  these  depending  upon 
the  feed  per  revolution  of  the  cutter,  and  the  depth  on  both  the 
feed  and  the  diameter  of  the  cutter.  The  ridges  are  more  prom- 
inent when  coarse  feeds  and  small  diameter  cutters  are  used. 
These  feed  marks  are  the  result  of  the  convex  path  of  the  cutting 
edge  and  the  slight  running  out  of  the  cutter,  which  is  inevitable 
in  all  rotary  cutters  with  a  number  of  teeth.  As  is  well  known 
to  those  familiar  with  milling  operations,  the  spacing  of  the 
marks  does  not  depend  on  the  number  of  teeth  in  the  cutter. 
Theoretically,  it  should  depend  on  this  number,  but  as  it  is  prac- 
tically impossible  to  get  a  cutter  which  will  run  absolutely  true 
with  the  axis  of  rotation,  only  one  mark  is  produced  for  each 
revolution,  and,  hence,  the  spacing  becomes  equal  to  the  feed 

137 


SPIRAL  GEARING 


per  revolution.  The  eccentricity  of  the  cutter  with  the  axis  of 
rotation  is,  therefore,  the  factor  which,  together  with  the  diam- 
eter of  the  cutter  and  the  feed  per  revolution,  determines  the 
quality  of  the  surface,  other  conditions  being  equal. 

The  depth  of  the  hollow  produced  by  the  high  side  of  the  re- 
volving cutter  is  equal  to  the  height  or  rise  of  a  circular  arc,  the 
radius  of  which  equals  the  radius  of  the  cutter,  and  the  chord 
of  which  equals  the  feed  per  revolution.  (See  Fig.  i.)  The 
length  of  the  chord  or  the  feed  per  revolution  may  be  expressed : 

F  =  2  X  V2HR-H2 

in  which  F  =  feed  per  revolution; 
H  =  height  of  arc; 
R  =  radius  of  cutter. 


Machinery 


Fig.  i.   Diagram  illustrating  the  Re-  Fig.  2.     Diagram  for  finding  Depth 
lation  between  Feed,  Diameter  of     of  Feed  Marks  on  Side  of  Tooth 
Cutter  and  Depth  of  Feed  Marks      cut  by  Milling  Cutter 

Since  F2  is  a  very  small  quantity,  it  may  be  discarded  in  the 
expression,  which  is  then  simplified  to  read: 

F  =  2  X 


in  which  D  =  diameter  of  cutter. 

Transposing  this  expression,  we  obtain  H  = 


—  —  , 


which  is 


an  approximately  correct  expression  of  the  depth  of  the  hollows 
produced  by  milling.  As  an  example,  take  an  8-pitch  rack 
cutter,  with  straight  rack-shaped  sides,  3  inches  in  diameter, 


GEAR  ROBBING 


139 


milling  with  a  feed  per  revolution  of  o.i  inch.  The  depth  of  the 
feed  marks  at  the  bottom  of  the  cut  will  be  equal  to : 

(        \2 

^          =  0.0008 ?>  inch. 
4X3 

The  working  surface  of  the  tooth,  however,  is  produced  by 
the  side  of  the  cutter,  as  illustrated  in  Fig.  2,^ and  the  depth  of 
the  feed  marks  is  normal  to  the  surface,  and  is  expressed  as: 

d  =  H  X  sin  a 

in  which  d  =  depth  of  the  feed  marks  on  the  side  of  the  tooth, 
and  a  the  angle  of  obliquity.  In  the  example  given,  the  depth 
d  would  equal  0.00021  inch,  for  a  i4§-degree  involute  tooth. 

The  depth  of  the  feed  marks  is  inversely  proportional  to  the 
diameter  of  the  cutter,  and  is,  therefore,  greater  at  the  point  of 


Fig.  3.    Angle  which 
limits  the  Feed 


Fig.  4.     Diagram  for  deducing  Formulas  for 
analyzing  Action  in  Gear  Hobbing  Machine 


the  tooth  than  at  the  root.  In  the  example  given  the  depth 
would  be  0.00025  inch  at  the  extreme  point  of  the  rack  tooth. 
It  is  thus  apparent  that  the  quality  of  the  surface  at  any  posi- 
tion along  the  tooth  from  the  root  to  the  point  depends  upon 
the  diameter  and  form  of  the  cutter  and  the  feed  per  revolution. 
In  Fig.  3  is  shown  the  outline  of  a  No.  6  standard  i4j-degree 
involute  gear  cutter.  This  outline,  at  the  point  close  to  the  end 
of  the  tooth  of  the  gear,  is  a  tangent  inclined  at  an  angle  of  45 
degrees,  as  indicated.  Hence,  the  depth  of  the  revolution  marks 
is: 

^°         X  sin  45°  =  0.000707  inch,  instead  of  0.00024  inch,  as 
4  X  2.5 

in  the  case  of  the  straight  rack  tooth.  It  is  evident  that  to  pro- 
duce an  equal  degree  of  finish  with  that  left  by  the  rack  cutter, 
the  feed  must  be  considerably  less  for  a  No.  6  involute  gear 


140  SPIRAL  GEARING 

cutter  than  for  the  rack  cutter.  In  Fig.  5  is  shown  the  full  range 
of  cutter  profiles  from  Nos.  i  to  8,  with  the  angle  of  the  tangent 
in  each  case  which  determines  the  quality  of  the  surface  under 
equal  conditions  of  feed  and  diameter  of  cutter. 

If  the  depth  of  the  feed  marks  is  used  as  the  determining  factor 
in  comparing  the  condition  of  the  surfaces  produced  by  a  series 
of  cutters,  it  is  evident  that  if  the  surface  produced  by  the  rack 
cutter  is  taken  as  a  standard,  the  feed  for  cutting  a  pinion  must 
be  considerably  less  than  the  feed  used  for  cutting  gears  with  a 
large  number  of  teeth.  In  fact,  if  a  rack  cutter  is  fed  o.ioo  inch 
per  revolution,  a  No.  8  standard  involute  gear  cutter  should 
not  be  fed  more  than  0.055  mcn  Per  revolution  to  produce  an 
equally  good  surface.  The  feed  is  proportional  to  the  square 


No.1         No.  2        No.  3 


Fig.  5.     Angles  limiting  theJTeed  for  14' ^-degree  Standard  Gear  Cutters 

root  of  the  reciprocal  of  the  sine  of  the  angle  of  the  limiting  tan- 
gent. 

If  we  assume  the  accuracy  of  the  surface  left  by  the  straight- 
sided  rack  cutter  as  equal  to  100  per  cent,  then  the  relative  feeds 
required  for  cutting  gears  with  any  formed  cutter  can  be  cal- 
culated. This  has  been  done,  and  the  results  are  shown  plotted 
in  curve  At  in  Fig.  6.  This  curve  is  based  on  an  equal  depth  of 
the  feed  marks  for  the  full  range  of  numbers  of  teeth  in  the  gears. 
If,  on  the  other  hand,  the  surfaces  left  by  the  cutter  for  a  given 
feed  per  revolution  are  compared,  the  depth  of  the  feed  marks 
will  vary  with  the  sine  of  the  angle  of  the  limiting  tangent,  and 
taking  the  straight-sided  rack  cutter  as  a  basis,  the  relative 
accuracy  of  the  surfaces  is  inversely  proportional  to  the  sine  of 
the  angle,  and  is  plotted  in  curve  B,  in  Fig.  6. 

Comparison  between  Surfaces  Produced  by  Milling  and  Rob- 
bing. —  A  relation  has  now  been  established  between  the  quality 
of  the  surface  and  the  permissible  feeds  for  cutters  for  cutting 
gears  with  any  number  of  teeth.  We  will  now  consider  the  con- 
dition of  the  surface  produced  by  a  hob  in  a  gear-hobbing  ma- 


GEAR  ROBBING 


141 


chine.  The  hob  is  made  with  straight-sided  rack-shaped  teeth 
and  with  sides  of  a  constant  angle,  and  is  used  to  produce  gears 
with  any  number  of  teeth.  We  may  therefore  assume  that  it  is 
cutting  under  the  conditions  governing  the  rack  cutter,  as  just 
explained,  the  surface  produced  being  considered  merely  as  a 
milled  surface.  If  this  assumption  be  correct,  then  the  quality 
of  the  surface  produced  by  a  hob,  whether  cutting  a  gear  of  twelve 
teeth  or  of  two  hundred  teeth,  will  be  the  same  for  a  given  feed, 


90 


70 


60 


50 


40 


30 


0?° 

CUTTER 


10  18  26  34  42  50  58  66  T4  82  90  98  106  114  122  130  138  146  154 164 172 

Machinery 


Fig.  6.     Diagrams  showing  the  Relation  between  Feed,  Finish,  and 
Number  of  Teeth  when  cutting  Gears  with  Formed  Gear  Cutters 

and  the  same  relation  exists  between  the  hob  and  any  formed 
cutter  that  exists  between  the  rack  cutter  and  any  formed  cutter; 
hence,  curves  A  and  B,  in  Fig.  6,  may  be  assumed  to  show  the 
permissible  feeds  and  the  quality  of  the  surfaces  produced  by 
formed  cutters  when  compared  with  the  surfaces  produced  by 
a  hob,  provided  the  surfaces  are  considered  merely  as  milled 
surfaces.  However,  a  condition  enters  in  the  case  of  the  hob 
which  has  no  equivalent  in  the  case  of  the  formed  milling  cutter, 
a*id  this  influences  the  condition  of  the  surface.  This  condition 


142 


SPIRAL   GEARING 


is  the  distortion  of  the  hob  teeth  in  hardening  which  causes  them 
to  mar  the  surface  of  the  tooth  by  "side  swiping,"  producing  a 
rough  surface.  The  eccentricity  of  the  hob  with  the  axis  of 
rotation  also  has  a  different  effect  on  the  surface  than  in  the  case 
of  a  formed  gear  cutter.  The  effect  is  shown  in  a  series  of  flats 
running  parallel  with  the  bottom  of  the  tooth,  if  excessive; 
if  the  eccentricity  is  small,  the  effect  will  merely  be  to  round  the 
top  of  the  tooth.  These  inaccuracies,  however,  can  be  taken  care 
of  in  a  number  of  ways. 

Comparison  of  Output.  —  For  reasons  not  connected  with  the 
quality  of  the  surface,  the  hob  may  be  worked  at  a  greater  cutting 

Comparison  of  Time  Required  for  Cutting  Gears  on  Automatic  Gear- 
cutting  Machines  and  Hobbing  Machines 


Automatic  Gear  Cutters 

Gear-hobbing  Machines 

Number  of 

Teeth 

Feed,  Inches 

Time,  Minutes 

Feed,  Inches 

Time,  Minutes 

32 

O.O22 

is 

0.050 

6-5 

31 

O.O2O 

19 

0.050 

9 

24 

0.024 

22 

0.050 

6 

17 

O.02O 

8 

0.050 

4 

17 

O.O2O 

17-5 

0.050 

5 

16 

O.OlS 

0.050 

7 

13 

0.013 

6.2S 

0.050 

6 

speed  and  feed  than  a  rotary  cutter,  when  cutting  from  the  solid, 
the  reason  being  due  to  the  generating  action  of  the  hob  which 
results  in  the  breaking  of  the  chips.  This  preserves  the  cutting 
edges  and  reduces  the  heating  effect  of  the  cut,  and  explains 
why  the  hob  may  give  such  good  results  as  compared  with  a 
rotary  cutter  in  the  matter  of  output.  It  is  possible  to  get  good 
results  in  the  general  run  of  work  in  the  hobbing  machine  in  one- 
third  to  one-half  of  the  time  required  in  an  automatic  gear  cutter. 
The  accompanying  table  gives  the  results  obtained  on  automobile 
transmission  gears  with  automatic  gear-cutting  machines  and 
hobbing  machines.  If  anything,  the  conditions  under  which 
the  comparisons  were  made  favored  the  automatic  machines. 
Here,  of  course,  spur  gears  are  considered,  but  the  relative  ad- 
vantages are  still  greater  in  the  case  of  spiral  gears.  The  speed 


GEAR  MOBBING  143 

of  the  cutter  in  all  cases  was  120  revolutions  per  minute,  except 
in  the  case  of  the  i3-tooth  pinion,  when  the  speed  was  raised  to 
160  R.P.M.  to  increase  the  output.  The  hob  was  run  at  a  speed 
of  105  R.P.M.  in  all  cases.  The  hob  and  cutters  were  of  practi- 
cally the  same  diameter.  The  results  were  obtained  in  producing 
an  ordinary  day's  work  and  clearly  indicate  the  advantage  of 
the  bobbing  process  over  the  milling  process^  when  the  quality 
of  the  tooth  surface  alone  is  considered,  on  the  basis  that  both 
processes  produce  a  milled  surface. 

The  Tooth  Outline.  —  Going  further  into  the  subject,  we  will 
take  up  the  question  of  the  tooth  outline.  The  tooth  of  a  gear 
milled  with  an  ordinary  milling  cutter  must  be,  or  at  least  is 
expected  to  be,  a  reproduction  of  the  outline  of  the  cutter,  and 
since  each  cutter  must  cover  a  wide  range  of  teeth,  the  outline 
is  not  theoretically  correct,  except  for  one  given  number  of  teeth 
in  the  range.  Theoretically  speaking,  the  outline  of  the  hobbed 
tooth  may  be  considered  as  a  series  of  tangents,  the  tooth  sur- 
face being  composed  of  a  series  of  flats  parallel  with  the  axis  of 
the  gear.  To  show  the  significance  of  these  flats,  assume,  for 
example,  that  a  gear  with  thirty-two  teeth  is  cut  with  a  stand- 
ard hob,  8  pitch,  3  inches  in  diameter,  having  twelve  flutes. 
The  length  of  the  portion  of  the  hob  that  generates  the  tooth 
surface  is  2  S  +  tan  a,  where  a  is  the  pressure  angle,  as  indicated 
in  Fig.  4.  The  number  of  teeth  following  in  the  generating 
path  is: 

/  2  S  \ 

( r-  circular  pitch )  X  number  of  gashes. 

Vtan  a  I 

In  this  case  the  generating  length  is  approximately  0.96  inch, 
and  there  are  thirty  teeth  in  the  generating  path.  The  flats  of 
those  parts  of  the  tooth  outline  which  each  of  the  hob  teeth  form 
vary  in  width  along  the  curves.  They  are  of  minimum  width 
at  the  base  line  and  of  maximum  width  at  the  point  of  the  tooth. 
The  width  of  the  flats  at  the  pitch  circle  is  proportional  to  the 
number  of  teeth  in  the  gear,  the  number  of  gashes  in  the  hob  and 
the  pressure  angle.  The  angle  /3,  to  the  left  in  Fig.  7,  which  is 
the  angle  between  each  flat,  is  proportional  to  the  number  of 


144 


SPIRAL  GEARING 


teeth  in  the  gear  and  the  number  of  gashes  in  the  hob.     In  the 
example  given  it  is: 


360  x 


3° 


=  0.91  degree,  or  55  minutes. 


The  Width  of  Flat  Produced.  —  The  width  a  of  the  flat  at  the 
pitch  line  is  equal  to  twice  the  tangent  of  one-half  0  times  the 


ROOT 
F  TOOTH 


Machinery 


Fig.  7.     Relative  Width  and  Position  of  Flats  produced  by  Gear 

Hobbing  Machines.     A  indicates  Feed  per  Each  Generating 

Tooth;  B,  Feed  per  Revolution  of  Blank 

length  of  the  pressure  line  between  the  point  of  tangency  with 
the  base  line  and  the  pitch  point,  and  is: 

a  =  2  tan  J  £  X  tan  14^°  X  2  =  0.0081  inch. 

This  is  not  a  flat  that  could  cause  serious  trouble.  As  in  the 
case  of  the  feed  marks,  it  is  not  the  width  of  the  flat  alone  that  is 
to  be  considered,  but  the  depth  must  be  taken  into  account; 
in  fact,  the  quality  of  the  surface  may  be  spoken  of  as  the  ratio 
of  the  depth  to  the  length  of  the  flat.  The  depth  of  the  flat  is 
the  rise  or  height  of  the  arc  of  the  involute  and  is  approximately 
proportional  to  the  versed  sine  of  the  angle  \  0,  and  with  the 
pitch  assumed  in  the  example  given  would  be  0.000015  inch.  It 
is  difficult  to  conceive  of  any  shock  caused  by  this  flat,  as  the 
gear  teeth  roll  over  each  other.  The  action  of  the  hob  and  gear 
in  relation  to  each  other  further  modifies  the  flat  by  giving  it  a 
crowning  or  convex  shape.  In  fact,  the  wider  the  flat  the  more 


GEAR  ROBBING  145 

it  is  crowned.  This  explains  the  fact  that  hobs  with  a  few 
gashes  produce  teeth  of  practically  as  good  shape  as  hobs  with  a 
large  number  of  gashes.  It  is  desirable,  therefore,  to  use  hobs 
with  as  few  gashes  as  possible,  because  from  a  practical  point  of 
view  the  errors  of  workmanship  and  those  caused  by  warping 
in  hardening  increase  with  the  number  of  flutes. 

A  peculiar  feature  of  the  hobbed  tooth  surface  is  shown  to  the 
right  in  Fig.  7,  which  illustrates  the  path  on  a  tooth  produced  by 
a  hob  in  one  revolution.  In  fact,  there  are  two  distinct  paths, 
the  first  starting  at  the  point  of  the  tooth  and  working  down  to 
the  base  line,  the  cutting  edges  of  the  hob  tooth  then  jumping 
to  the  root  of  the  tooth  and  working  up  to  the  base  line,  pro- 
ducing the  zigzag  path  shown. 

Summary  of  the  Preceding  Comparative  Study.  —  That  the 
flats  so  commonly  seen  in  the  results  obtained  from  the  hobbing 
machine  are  not  due  to  any  faults  of  the  process  that  cannot  be 
corrected,  but  are  due  to  either  carelessness  on  the  part  of  the 
operator  in  setting  up  the  machine  without  proper  support  to 
the  work,  or  to  the  poor  condition  of  the  hob  or  machine,  and 
that  nearly  all  cases  of  flats  can  be  overcome  by  the  use  of  a 
proper  hob,  may  be  assumed  as  a  statement  of  facts.  When 
the  hobbing  machine  will  not  give  good  results,  the  hob  is  in 
nearly  all  cases  at  fault.  If  a  gear  is  produced  that  bears  hard 
on  the  point  of  the  teeth,  has  a  flat  at  the  pitch  line  or  at  any  point 
along  the  face  of  the  tooth,  do  not  think  that  the  process  is  faulty 
in  theory,  or  that  the  machine  is  not  properly  adjusted,  or  that 
the  strain  of  the  cut  is  causing  undue  torsion  in  the  shafts,  or 
that  there  is  backlash  between  the  gears  in  the  train  connecting 
the  work  and  the  hob ;  these  things  are  not  as  likely  to  cause  the 
trouble  as  is  a  faulty  hob. 

After  an  experience  covering  all  makes  of  hobbing  machines, 
the  author  has  come  to  the  conclusion  that  the  real  cause  of  the 
trouble  in  nearly  every  case  is  a  faulty  hob.  Machine  after 
machine  has  been  taken  apart,  overhauled  and  readjusted,  and 
yet  no  better  results  have  been  obtained  until  a  new  and  better 
hob  was  produced.  The  faults  usually  met  with  in  hobs  will  be 
referred  to  in  the  following,  together  with  the  means  for  getting 


146  SPIRAL  GEARING 

the  hob  into  a  good  working  condition.  It  is  not  desired  in  any 
way  to  disparage  the  formed  cutter  process  in  favor  of  the  hobbing 
process,  but  simply  to  state  the  facts  as  they  appear.  In  every 
case,  practice  seems  to  substantiate  the  conclusions  arrived  at. 

Hobs  for  Spur  and  Spiral  Gears.  —  The  success  of  the  hobbing 
process  for  cutting  teeth  in  spur  and  spiral  gears  depends,  as 
stated,  more  upon  the  hob  than  upon  the  machine,  and  at  the 
present  stage  of  development  the  hob  is  the  limiting  factor  in 
the  quality  of  the  product.  It  is  well  known  that  hobs  at  the 
present  time  are  far  from  being  standardized,  and  that  the  product 
will  not  be  interchangeable  if  the  hob  of  one  maker  is  substituted 
for  that  of  another;  in  fact,  the  using  of  two  hobs  from  the  same 
maker  successively  will  sometimes  result  in  the  production  of 
gears  which  will  not  interchange  or  run  smoothly.  This  is  not  a 
fault  of  the  hobbing  process,  but  is  due  to  the  fact  that  the  cutter 
manufacturers  have  not  given  the  question  of  hobs  the  study  it 
requires.  This  is  also  the  reason  why  there  are  so  many  com- 
plaints about  the  hobbing  machine.  Nevertheless,  with  the 
proper  hob  the  hobbing  machine  is  the  quickest  method  of  machin- 
ing gears  that  has  ever  been  devised.  Its  advantage  lies  in  the 
continuous  action  and  in  the  simplicity  of  its  mechanism.  There 
is  no  machine  for  producing  the  teeth  of  spur  gears  that  can  be 
constructed  with  a  simpler  mechanism,  and  even  machines  using 
rotary  cutters  are  more  complicated  if  automatic. 

Hobs  with  Few  Teeth  Give  Best  Results.  —  The  ideal  form 
of  hob,  theoretically  speaking,  would  be  one  that  had  an  infinite 
number  of  cutting  teeth.  In  practice,  however,  a  seemingly 
contradictory  result  is  obtained,  as  hobs  with  comparatively  few 
teeth  give  the  best  results.  The  reasons  for  this  are  due  to 
purely  practical  considerations.  Strictly  speaking,  a  theoretical 
tooth  curve  is  no  more  possible  when  the  tooth  is  produced  by 
the  hobbing  process  than  when  produced  by  the  shaper  or  planer 
type  of  generator,  but  for  all  practical  purposes,  the  curve  gen- 
erated under  proper  working  conditions  is  so  nearly  correct  as 
to  be  classed  as  a  theoretical  curve.  If  this  result  is  not  often 
met  with  under  ordinary  working  conditions,  it  is  due  to  the  fact 
that  the  hob  is  not  as  good  as  present  practice  is  able  to  make  it. 


GEAR  ROBBING 


147 


Causes  of  Defects  in  Hobbed  Gears.  —  In  order  to  obtain,  as 
far  as  is  theoretically  possible,  a  proper  curve  and  not  a  series  of 
flat  surfaces,  the  teeth  of  the  hob  must  follow  in  a  true  helical 
path.  In  ninety-nine  cases  out  of  one  hundred  the  hob  is  at 
fault  when  a  series  of  flats  is  obtained  instead  of  a  smooth  curved 
tooth  face.  It  is  the  deviation  of  the  teeth  of  the  hob  from  the 
helical  path  that  is  at  the  root  of  most  hobbing  machine  troubles. 
There  are  several  causes  for  the  teeth  being  out  of  the  helical 
path:  The  trouble  may  have  originated  in  the  relieving  or  form- 
ing of  the  teeth ;  the  machine  on  which  this  work  has  been  done 


Machinery 


Figs.  8  and  9.     Distortion  of  Hob  Teeth  and  its  Effect 

may  have  been  too  light  in  construction,  so  that  the  tool  has  not 
been  held  properly  to  its  work,  and  has  sprung  to  one  side  or 
another  causing  thick  and  thin  teeth  in  the  hob;  a  hard  spot 
may  have  been  encountered  causing  the  tool  to  spring;  the 
gashes  may  not  have  been  properly  spaced,  or  there  may  have 
been  an  error  in  the  gears  on  the  relieving  lathe  influencing  the 
form;  the  hob  may  also  have  been  distorted  in  hardening;  it 
may  have  been  improperly  handled  in  the  fire  or  bath,  or  it  may 
have  been  so  proportioned  that  it  could  not  heat  or  cool  uniformly; 
the  grinding  after  hardening  may  be  at  fault;  the  hole  may  not 
have  been  ground  concentric  with  the  form,  thus  causing  the 
teeth  on  one  side  of  the  hob  to  cut  deeper  than  on  the  other. 


148  SPIRAL  GEARING 

Any  one  or  a  combination  of  several  of  these  conditions  may 
have  thrown  the  teeth  out  of  the  true  helical  path. 

Fig.  8  illustrates  the  difficulty  of  thick  and  thin  teeth.  The 
tooth  A  is  too  thick  and  B  is  too  thin,  the  threading  tool  having 
sprung  over  from  A  and  gouged  into  B.  Fig.  8  also  shows  a 
developed  layout  of  the  hob.  At  C  is  shown  the  effect  that 
thick  and  thin  teeth  may  have  on  the  tooth  being  cut  in  the 
gear  —  that  of  producing  a  flat  on  the  tooth.  This  flat  may 
appear  on  either  side  of  the  tooth  and  at  almost  any  point  from 
the  root  to  the  top,  depending  upon  whether  the  particular  hob 
tooth  happens  to  come  central  with  the  gear  or  not.  If  it  does 
come  central  or  nearly  so,  it  will  cause  the  hob  to  cut  thin  teeth 
in  the  blank.  The  only  practical  method  to  make  a  hob  of  this 
kind  fit  for  use  is  to  have  it  re-formed. 

In  Fig.  9  is  shown  at  A  the  result  of  unequal  spacing  of  the 
teeth  around  the  blank.  Owing  to  the  nature  of  the  relief,  the 
unequal  spacing  will  cause  the  top  of  the  teeth  to  be  at  different 
distances  from  the  axis  of  the  hob.  This  would  produce  a  series 
of  flats  on  the  gear  tooth.  One  result  of  distortion  in  hardening 
is  shown  at  B}  Fig.  9,  where  the  tooth  is  canted  over  to  one  side 
so  that  one  corner  is  out  of  the  helical  path.  This  defect  also 
produces  a  flat  and  shows  a  peculiar  under-cutting  which  at  first 
is  difficult  to  account  for.  Sometimes  a  tooth  will  distort  under 
the  effects  of  the  fire  in  the  manner  indicated  at  D,  Fig.  9. 

These  defects  may  be  avoided  by  proper  care  and  by  having 
the  steel  in  good  condition  before  forming.  The  blank  should 
be  roughed  out,  bored,  threaded,  gashed  and  then  annealed  before 
finish-forming  and  hardening.  The  annealing  relieves  the  stresses 
in  the  steer  due  to  the  rolling  process. 

The  proportions  of  the  hob  have  a  direct  effect  on  distortion 
in  hardening.  This  is  especially  noticeable  in  hobs  of  large 
diameter  for  fine  pitches.  Fig.  10  shows  the  results  obtained  in 
hardening  a  4-inch  hob,  10  pitch,  with  ij-inch  hole.  There  is 
a  bulging  or  crowning  of  the  teeth  at  A .  This  is  accounted  for 
by  the  fact  that  the  mass  of  metal  at  B  does  not  cool  as  quickly 
as  that  at  the  ends.  Consequently,  when  the  hob  is  quenched, 
the  ends  and  outer  shell  cool  most  quickly  and  become  set,  pre- 


GEAR  HOBBING 


149 


venting  the  mass  at  B  from  contracting  as  it  would  if  it  could 
come  in  direct  contact  with  the  cold  bath  and  cool  off  as  quickly 
as  the  rest  of  the  metal.  The  effect  of  this  distortion  on  the 
shape  of  the  gear  teeth  is  indicated  in  Fig.  n,  where  the  tooth 
A  is  unsymmetrical  in  shape  due  to  the  fact  that  the  teeth  near 
the  center  of  the  hob  cut  deeper  into  the  blank,  under-cutting 
the  tooth  on  one  side  and  thinning  the  point.  This  effect  is  pro- 
duced when  the  gear  is  centered  near  the  ends  of  the  hob.  If 
the  gear  is  centered  midway  of  the  length  of  the  hob,  the  tooth 


Machinery 


Figs.  10  and  n. 


Distortion  of  Hobs  and  Result  on  the  Shape  of 
the  Teeth 


shape  produced  is  as  shown  at  B,  Fig.  n.  This  tooth  is  thick 
at  the  point  and  under-cut  at  the  root. 

A  hob  in  this  condition  makes  it  impossible  to  obtain  quiet 
running  gears.  In  this  case,  it  would  be  useless  to  anneal  and 
re-form  the  hob,  as  the  same  results  would  be  certain  to  be  met 
with  again,  on  account  of  the  proportions  of  the  hob.  Hence, 
defects  of  this  kind  are  practically  impossible  to  correct,  and  the 
hob  should  either  be  entirely  remade  or  discarded. 

Figs.  12  and  13  show  in  an  exaggerated  manner  two  common 
defects  due  to  poor  workmanship.  In  Fig.  12  the  hole  is  ground 
out  of  true  with  the  outside  of  the  tooth  form.  The  hole  may  run 
either  parallel  with  the  true  axis  of  the  hob,  or  it  may  run  at  an 
angle  to  it,  as  seen  in  Fig.  13. 

The  effect  of  the  first  condition  is  to  produce  a  tooth  shaped 
like  that  shown  by  the  full  lines  at  B  in  Fig.  n,  and  the  effect 


SPIRAL  GEARING 


of  that  in  Fig.  13  is  about  the  same,  except  that  the  hob  will 
cut  thin  teeth  when  cutting  to  full  depth.  Gears  cut  by  either 
hob  will  lock  with  meshing  gears,  and  instead  of  smooth  rolling, 
the  action  will  be  jerky  and  intermittent. 

Gashes  which  originally  were  equally  spaced  may  have  become 
unequally  spaced  by  having  more  ground  off  the  face  of  some 
teeth  than  of  others.  The  greater  the  amount  of  relief,  the  more 
particular  one  must  be  in  having  the  gashes  equally  spaced. 

Grinding  to  Correct  Hob  Defects.  —  These  various  faults  may 
be  corrected  to  a  greater  or  less  extent  in  the  following  manner: 


CENTER  LINE 
OF  FORM  " 


CENTER  LINE/ 
OF  HOLE 


Machinery 


Figs.  12  and  13.     Hobs  with  the  Center  Hole  out  of  True  with  the 
Outside  of  the  Tooth  Form 

Place  the  hob  on  a  true  arbor  and  grind  the  outside  as  a  shaft 
would  be  ground;  touch  all  of  the  teeth  just  enough  so  that  the 
faintest  marks  of  the  wheel  can  be  seen  on  the  tops.  The  teeth 
that  are  protruding  and  would  cause  trouble  will,  of  course,  show 
a  wide  ground  land,  while  on  those  that  are  low,  the  land  will 
be  hardly  visible.  Now  grind  the  face  of  each  tooth  back  until 
the  land  on  each  is  equal.  This  will  bring  all  the  teeth  to  the 
same  height  and  the  form  will  run  true  with  the  hole.  To  keep 
the  hob  in  condition  so  that  it  will  not  be  spoiled  at  the  first  re- 
sharpening,  grind  the  backs  of  the  teeth,  using  the  face  as  a 
finger-guide,  the  same  as  when  sharpening  milling  cutters,  so  as 
to  remove  enough  from  the  back  of  each  tooth  to  make  the  dis- 
tance AS,  Fig.  12,  the  same  on  all  the  teeth.  Then,  when  shar- 
pening the  teeth  in  the  future,  use  the  back  of  the  tooth  as  a 
finger-guide.  If  care  is  taken,  the  hob  will  then  cut  good  gears 
as  long  as  it  lasts.  It  is  poor  practice  to  use  the  index  head  when 


GEAR  ROBBING  151 

sharpening  hobs,  because  the  form  is  never  absolutely  true  with 
the  hole,  and  unless  the  hob  has  been  prepared  as  just  described, 
there  is  no  reliable  way  to  sharpen  it.  If  the  hob,  after  having 
been  prepared  as  described,  is  sharpened  on  centers  by  means 
of  indexing,  it  will  be  brought  back  to  the  original  condition. 

The  defect  shown  in  Fig.  13  is  corrected  in  the  same  manner. 
The  gash  when  so  ground  will  not  be  parallel  with  the  axis  in  a 
straight-fluted  hob,  nor  will  it  be  at  an  exact  right  angle  with 
the  thread  helix  in  a  spiral-fluted  hob,  because  the  teeth  at  the 
right-hand  end  are  high  while  those  at  the  left-hand  end  are 
low,  and  the  amount  that  must  be  ground  off  the  faces  of  the 
hob  teeth  will  be  greater  at  one  end  than  at  the  other.  The  angle 
will  be  slight,  however,  and  of  no  consequence. 

Shape  of  Hob  Teeth.  —  The  first  thing  that  is  questioned  when 
a  hob  does  not  produce  smooth  running  gears  is  the  shape  of  the 
hob  tooth.  The  poor  bearing  obtained  when  rolling  two  gears 
together  would,  in  many  cases,  seem  to  indicate  that  the  hob 
tooth  was  of  improper  shape,  but  in  nearly  every  case  the  trouble 
is  the  result  of  one  or  more  of  the  defects  already  pointed  out. 

Theoretically,  the  shape  of  the  hob  tooth  should  be  that  of 
a  rack  tooth  with  perfectly  straight  sides.  This  shape  will  cut 
good  gears  from  thirty  teeth  and  up,  in  the  i^-degree  involute 
system,  but  gears  under  thirty  teeth  will  have  a  reduced  bearing 
surface  as  a  result  of  under-cutting  near  the  base  circle,  which 
increases  as  the  number  of  teeth  grows  smaller.  The  shape  pro- 
duced by  such  a  hob,  if  mechanically  perfect,  would  be  a  correct 
involute,  and  the  gears  should  interchange  without  difficulty. 
In  order  that  the  beginning  of  contact,  however,  may  take  place 
without  jar,  the  points  of  the  teeth  should  be  relieved  or  thinned, 
so  that  the  contact  takes  place  gradually,  instead  of  with  full 
pressure.  This  is  accomplished  by  making  the  hob  tooth  thicker 
at  the  root,  starting  at  a  point  considerably  below  the  pitch  line. 
This  is  illustrated  in  the  upper  portion  of  Fig.  14,  which  shows 
the  standard  shape  adopted  by  the  Barber-Colman  Co.  The 
shape  of  the  tooth  produced  is  also  shown.  The  full  lines  show 
the  shape  generated,  and  the  dotted,  the  lines  of  the  true  involute. 
The  amount  removed  from  the  points  is  greater  on  large  gears 


152 


SPIRAL  GEARING 


and  less  on  small  pinions,  where  the  length  of  contact  is  none  too 
great  even  with  a  full-shaped  tooth,  and  where  any  great  reduc- 
tion must  be  avoided.  This  shape  of  hob  tooth  does  not,  how- 
ever, reduce  under-cutting  on  small  pinions.  The  fact  that 
bobbed  gears  have  under-cut  teeth  in  small  pinions,  while  those 
cut  with  rotary  cutters  have  radial  flanks  with  the  curve  above 
the  pitch  line  corrected  to  mesh  with  them  is  the  reason  why 


Fig.  14.    Forms  of  Hob  Teeth  and  Gear  Teeth  Produced 

hobbed  gears  and  those  cut  with  rotary  cutters  will  not  inter- 
change. 

In  the  lower  part  of  Fig.  14  is  shown  a  hob  tooth  shape  which 
will  produce  teeth  in  pinions  without  under-cutting,  the  teeth, 
instead,  having  a  modified  radial  flank.  The  radii  of  the  cor- 
rection curves  are  such  that  the  gear  tooth  will  be  slightly  thin 
at  the  point  to  allow  an  easy  approach  of  contact.  The  shape 
shown  is  approximately  that  which  will  be  produced  on  hobs, 
the  teeth  of  which  are  generated  from  the  shape  of  a  gear  tooth 
cut  with  a  rotary  cutter,  which  it  is  desired  to  reproduce. 


GEAR  ROBBING 


153 


Diameters  of  Hobs.  —  The  diameters  of  hobs  is  a  subject 
which  has  been  much  discussed.  Many  favor  large  hobs  because 
the  larger  the  hob  the  greater  the  number  of  teeth  obtainable. 
This,  however,  has  already  been  shown  to  be  a  fault,  because 
the  greater  is  the  possible  chance  of  some  of  the  teeth  being  dis- 
torted. For  the  same  feed,  the  output  of  a  small  hob  is  greater, 
because  of  being  inversely  proportional  to  the  -diameter.  The 
number  of  teeth  cut  is  directly  proportional  to  the  number  of 
revolutions  per  minute  of  the  hob.  The  number  of  revolutions 
depends  on  the  surface  speed  of  the  hob;  therefore,  the  small 
hob  will  produce  more  gears  at  a  given  surface  speed. 


Machinery 


Fig.  15.     Illustrating  Effect  of  Feed  in  Robbing 

It  may  be  argued  that,  on  account  of  the  large  diameter,  the 
large  hob  can  be  given  a  greater  feed  per  revolution  of  the  blank 
than  the  smaller  hob,  for  a  given  quality  of  tooth  surface.  This 
argument  is  analyzed  in  Fig.  15.  Let  R  be  the  radius  of  the  hob 
and  F  the  feed  of  the  hob  per  revolution  of  the  blank.  Then  B 
may  be  called  the  rise  of  feed  arc. 

Since  the  surface  of  the  tooth  is  produced  by  the  side,  the 
actual  depth  of  the  feed  marks  is  D,  which  depends  on  the  angle 
of  the  side  of  the  tooth,  the  depth  being  greater  for  a  2o-degree 
tooth  than  it  would  be  for  a  i^-degree  tooth  for  the  same  amount 


154 


SPIRAL  GEARING 


of  feed.     The  relations  between  F,  R,  and  B  may  be  expressed 
as  follows: 

F  =  2  V2RB-B2 

Since  B  is  a  very  small  fractional  quantity,  B2  would  be  much 
smaller  and  can,  therefore,  be  disregarded,  giving  the  very  simple 
approximate  formula  F  =  2  V2  RB.  A  rise  of  o.ooi  inch 
would  mean  a  depth  D  of  about  0.00025  inch  on  a  i4^-degree 
tooth.  The  allowable  feed  is  0.126  inch  for  a  4-inch  hob  and 
0.108  inch  for  a  3-inch  hob  for  a  o.ooi  inch  rise.  The  curve  in 


0.150 
0.140 
0.130 
0.120 
g  0.110 
0.100 
0.090 
0.030 
0.070 

150 
140 
130 

120  jj- 
o 

-j 

100  g 
90 
80 

70 
tnery 

\ 

/ 

/ 

\ 

/ 

V 

/ 

\ 

2 

t 

\ 

/ 

\ 

/ 

\ 

/ 

\ 

/ 

\ 

/ 

\ 

x 

/ 

\ 

x^ 

/ 

s 

\ 

/ 

^ 

s 

/ 

2345 
DIAMETER  OF  HOB 

2345 

DIAMETER  OF  HOB 
Mach 

Fig.  1 6.  Diagram  showing  Com-  Fig.  17.  Diagram  showing  Com- 
parative Feeds  for  Hobs  of  Various  parative  Output  of  Hobs  of  Various 
Diameters  Diameters  based  on  3-inch  Hob 

Fig.  1 6  shows  the  feeds  for  this  rise  for  various  hob  diameters. 
Fig.  17  shows  a  curve  based  on  a  3-inch  hob  that  shows  the  com- 
parative output  for  an  equal  rise.  This  curve  shows  that  the 
smaller  hob  is  superior  in  matter  of  production. 

The  larger  hobs  are  also  more  liable  to  distortion  in  hardening 
and  they  do  not  clear  themselves  as  well  as  the  smaller  ones  when 
cutting;  consequently,  they  need  a  greater  amount  of  relief. 
Large  hobs  also  require  a  greater  over-run  of  feed  at  the  start 
of  the  cut.  When  cutting  spiral  gears  of  large  angles  this  greatly 


GEAR  ROBBING  155 

reduces  the  output,  as  the  greater  amount  of  feed  required  before 
the  hob  enters  to  full  depth  in  the  gear  is  a  pure  waste. 

The  question  of  whether  the  gashes  or  flutes  should  be  parallel 
with  the  axis  or  at  right  angles  to  the  thread  helix  has  two  sides. 
From  a  practical  point  of  view,  it  appears  to  make  very  little 
difference  in  the  results  obtained  in  hobs  of  small  pitch  and  angle 
of  thread.  In  hobs  of  coarse  pitch,  however,  the  gashes  should 
undoubtedly  be  normal  to  the  thread.  The  effect  of  the  straight 
gash  is  noticed  when  cutting  steel,  in  that  it  is  difficult  to  obtain 
a  smooth  surface  on  one  side  of  the  tooth,  especially  when  cutting 
gears  •  coarser  than  10  pitch.  What  has  been  said  in  the  fore- 
going, however,  applies  equally  to  straight  and  spirally  fluted 
hobs. 


CHAPTER  VI 
CALCULATING  THE  DIMENSIONS   OF  WORM   GEARING 

THE  present  chapter  contains  a  compilation  of  rules  for  the 
calculation  of  the  dimensions  of  worm  gearing,  expressed  with 
as  much  simplicity  and  clearness  as  possible.  No  attempt  has 
been  made  to  give  rules  for  estimating  the  strength  or  durability 
of  worm  gearing,  although  the  question  of  durability,  especially, 
is  the  determining  factor  in  the  design  of  worm  gearing.  If  the 
worm  and  wheel  are  so  proportioned  as  to  have  a  reasonably 


--H 


SINGLE  THREAD 


DOUBLE  THREAD 


TRIPLE  THREAD 

Machinery 


Fig.  i.     Diagram  showing  Relation  between  Lead  and  Pitch 

long  life  under  normal  working  conditions,  it  may  be  taken  for 
granted  that  the  teeth  are  strong  enough  for  the  load  they  have 
to  bear.  No  simple  rules  have  ever  been  proposed  for  propor- 
tioning worm  gearing  to  suit  the  service  it  is  designed  for.  Judg- 
ment and  experience  are  about  the  only  factors  the  designer  has 
for  guidance.  In  Europe  a  number  of  builders  are  regularly 
manufacturing  worm  drives,  guaranteed  for  a  given  horsepower 
at  a  given  speed.  Reference  to  the  durability  and  power  trans- 
mitting properties  of  worm  gearing  will  be  made  in  a  following 
chapter. 

156 


RULES  AND   FORMULAS 


157 


Definitions  and  Rules  for  Dimensions  of  the  Worm.  —  In 

giving  names  to  the  dimensions  of  the  worm,  there  is  one  point 
in  which  there  is  sometimes  confusion.  This  relates  to  the  dis- 
tinction between  the  terms  "pitch"  and  "lead."  In  the  follow- 
ing we  will  adhere  to  the  nomenclature  indicated* in  Fig.  i. 
Here  are  shown  three  worms,  the  first  single-threaded,  the  sec- 
ond double-threaded,  and  the  last  triple-threaded.  As  shown, 
the  word  "lead"  is  assumed  to  mean  the  distance  which  a  given 
thread  advances  in  one  revolution  of  the  worm,  while  by  "pitch, " 
or  more  strictly,  "linear  pitch,"  we  mean  the  distance  between 
the  centers  of  two  adjacent  threads.  As  may  be  clearly  seen, 
the  lead  and  linear  pitch  are  equal  for  a  single-threaded  worm. 


DOUBLE  THREAD 


Fig.  2.     Nomenclature  used  for  Worm  Dimensions 

For  a  double-threaded  worm  the  lead  is  twice  the  linear  pitch, 
and  for  a  triple-threaded  worm  it  is  three  times  the  linear 
pitch.  From  this  we  have: 

Rule  i.  To  find  the  lead  of  a  worm,  multiply  the  linear  pitch 
by  the  number  of  threads. 

It  is  understood,  of  course,  that  by  the  number  of  threads  is 
meant  not  the  number  of  threads  per  inch,  but  the  number  of 
threads  in  the  whole  worm  —  one,  if  it  is  single-threaded,  four, 
if  it  is  quadruple-threaded,  etc.  Rule  i  may  be  transposed  to 
read  as  follows: 

Rule  2.  To  find  the  linear  pitch  of  a  worm,  divide  the  lead  by 
the  number  of  threads. 


158  WORM  GEARING 

The  standard  form  of  worm  thread,  measured  in  an  axial  sec- 
tion, as  shown  in  Fig.  2,  has  the  same  dimensions  as  the  standard 
form  of  involute  rack  tooth  of  the  same  linear  or  circular  pitch. 
It  is  not  of  exactly  the  same  shape,  however,  not  being  rounded 
at  the  top,  nor  provided  with  fillets.  The  thread  is  cut  with  a 
straight-sided  tool,  having  a  square,  flat  end.  The  sides  have 
an  inclination  with  each  other  of  29  degrees,  or  14 J  degrees  with 
the  center  line.  The  following  rules  give  the  dimensions  of  the 
teeth  in  an  axial  section  for  various  linear  pitches.  For  nomen- 
clature, see  Fig.  2. 

Rule  3.  To  find  the  whole  depth  of  the  worm  tooth,  multiply 
the  linear  pitch  by  0.6866. 

Rule  4.  To  find  the  width  of  the  thread  tool  at  the  end,  mul- 
tiply the  linear  pitch  by  0.31. 

Rule  5.  To  find  the  addendum  or  height  of  worm  tooth  above 
the  pitch  line,  multiply  the  linear  pitch  by  0.3183. 

Rule  6.  To  find  the  outside  diameter  of  the  worm,  add 
together  the  pitch  diameter  and  twice  the  addendum. 

Rule  7.  To  find  the  pitch  diameter  of  the  worm,  subtract 
twice  the  addendum  from  the  outside  diameter. 

Rule  8.  To  find  the  bottom  diameter  of  the  worm,  subtract 
twice  the  whole  depth  of  tooth  from  the  outside  diameter. 

Rule  9.  To  find  the  helix  angle  of  the  worm  and  the  gashing 
angle  of  the  worm-wheel  tooth,  multiply  the  pitch  diameter  of 
the  worm  by  3.1416,  and  divide  the  product  by  the  lead;  the 
quotient  is  the  cotangent  of  the  tooth  angle  of  the  worm. 

Rules  for  Dimensioning  the  Worm-wheel.  —  The  dimensions 
of  the  worm-wheel,  named  in  the  diagram  shown  in  Fig.  3,  are 
derived  from  the  number  of  teeth  determined  upon  for  it,  and 
the  dimensions  of  the  worm  with  which  it  is  to  mesh.  The  fol- 
lowing rules  may  be  used: 

Rule  10.  To  find  the  pitch  diameter  of  the  worm-wheel,  mul- 
tiply the  number  of  teeth  in  the  wheel  by  the  linear  pitch  of  the 
worm,  and  divide  the  product  by  3.1416. 

Rule  ii.  To  find  the  throat  diameter  of  the  worm-wheel, 
add  twice  the  addendum  of  the  worm  tooth  to  the  pitch  diameter 
of  the  worm  wheel. 


RULES  AND   FORMULAS 


Rule  12.  To  find  the  radius  of  curvature  of  the  worm-wheel 
throat,  subtract  twice  the  addendum  of  the  worm  tooth  from 
half  the  outside  diameter  of  the  worm. 

The  face  angle  of  the  wheel  is  arbitrarily  selected;  60  degrees 
is  a  good  angle,  but  it  may  be  made  as  high  as  80  or  even  90  de- 
grees, though  there  is  little  advantage  in  carrying  the  gear  around 


•^RADIUS  OF  CURVATURE  OF  THROAT=U 
:E  ANGLE  =  ( 


H  X 


Machinery 


Fig.  3.    Nomenclature  used  for  Worm-gear  Dimensions 

so  great  a  portion  of  the  circumference  of  the  worm,  especially 
in  steep  pitches. 

Rule  13.  To  find  the  diameter  of  the  worm-wheel  to  sharp 
corners,  multiply  the  radius  of  curvature  of  the  throat  by  the 
cosine  of  half  the  face  angle,  subtract  this  quantity  from 
the  radius  of  curvature  of  throat,  multiply  the  remainder  by 
2,  and  add  the  product  to  the  throat  diameter  of  the  worm- 
wheel. 


160  WORM   GEARING 

If  the  sharp  corners  are  flattened  a  trifle  at  the  tops,  as  shown 
in  Figs.  3  and  5,  this  dimension  need  not  be  figured,  "trimmed 
diameter"  being  easily  scaled  from  an  accurate  drawing  of  the 
gear. 

There  is  a  simple  rule  which,  rightly  understood,  may  be  used 
for  obtaining  the  velocity  ratio  of  a  pair  of  gears  of  any  form, 
whether  spur,  spiral,  bevel  or  worm.  The  number  of  teeth  of 
the  driven  gear,  divided  by  the  number  of  teeth  of  the  driver, 
will  give  the  velocity  ratio.  For  worm  gearing  this  rule  takes 
the  following  form: 

Rule  14.  To  find  the  velocity  ratio  of  a  worm  and  worm-wheel, 
divide  the  number  of  teeth  in  the  wheel  by  the  number  of  threads 
in  the  worm. 

Be  sure  that  the  proper  meaning  is  attached  to  the  phrase 
"number  of  threads"  as  explained  before  under  Rule  i.  The 
revolutions  per  minute  of  the  worm,  divided  by  the  velocity 
ratio,  gives  the  revolutions  per  minute  of  the  worm-wheel. 

Rule  15.  To  find  the  distance  between  the  center  of  the  worm- 
wheel  and  the  center  of  the  worm,  add  together  the  pitch  diam- 
eter of  the  worm  and  the  pitch  diameter  of  the  worm-wheel, 
and  divide  the.  sum  by  2. 

Rule  1 6.  To  find  the  pitch  diameter  of  the  worm,  subtract 
the  pitch  diameter  of  the  worm-wheel  from  twice  the  center 
distance. 

The  worm  should  be  long  enough  to  allow  the  wheel  to  act  on 
it  as  far  as  it  will.  The  length  of  the  worm  required  for  this 
may  be  scaled  from  a  carefully-made  drawing,  or  it  may  be  cal- 
culated by  the  following  rule: 

Rule  17.  To  find  the  minimum  length  of  worm  for  complete 
action  with  the  worm-wheel,  subtract  four  times  the  addendum 
of  the  worm  thread  from  the  throat  diameter  of  the  wheel,  square 
the  remainder,  and  subtract  the  result  from  the  square  of  the 
throat  diameter  of  the  wheel.  The  square  root  of  the  result  is 
the  minimum  length  of  worm  advisable. 

The  length  of  the  worm  should  ordinarily  be  longer  than  the 
dimension  thus  found.  Hobs,  particularly,  should  be  long  enough 
for  the  largest  wheels  they  are  ever  likely  to  be  called  upon  to  cut. 


RULES  AND  FORMULAS  161 

Departures  from  the  Foregoing  Rules.  —  The  throat  diameter 
of  the  wheel  and  the  center  distance  may  have  to  be  altered  in 
some  cases  from  the  figures  given  by  the  preceding  rules.  If 
worm-wheels  with  small  numbers  of  teeth  are  made  to  the  dimen- 
sions given,  it  will  be  found  that  the  flanks  of  the  teeth  will  be 
partly  cut  away  by  the  tops  of  the  hob  teeth,  so  that  the  full 
bearing  area  is  not  available.  The  matter  becomes  serious  when 
there  are  less  than  25  or  30  teeth  in  the  worm-wheel.  One 
method  of  avoiding  this  under-cutting  is  to  increase  the  throat 
diameter  of  the  wheel  blank  in  accordance  with  the  following 
rule:  To  obtain  the  throat  diameter,  multiply  the  pitch  diameter 
of  the  wheel  by  0.937  and  add  to  the  product  4  times  the  adden- 
dum of  the  worm-wheel  tooth.  This  diameter  can  also  be  ob- 
tained as  follows:  Multiply  the  product  of  the  circular  pitch 
and  number  of  teeth  in  the  worm-wheel  by  0.298;  then  add  1.273 
times  the  circular  pitch.  If  it  is  necessary  to  keep  the  original 
center-to-center  distance,  the  outside  diameter  of  the  worm 
must  be  reduced  the  same  amount  that  the  throat  diameter  is 
increased.  When  turning  blanks,  it  is  the  general  practice  to 
simply  reduce  the  central  part  of  the  throat  to  the  required  diam- 
eter, the  remainder  being  left  somewhat  over  size  so  that  the  tops 
of  the  teeth  will  be  finished  to  the  proper  radius  by  the  hob. 

On  the  other  hand,  some  designers  claim  to  get  better  results 
in  efficiency  and  durability  by  making  the  throat  diameter  of  the 
worm-wheel  smaller  than  standard,  where  it  is  possible  to  do  so 
without  too  much  under-cutting.  Under  no  conditions,  however, 
should  the  throat  diameter  ever  be  made  so  small  as  to  produce 
more  interference  than  is  met  with  in  a  standard  25-tooth  worm- 
wheel. 

Two  Applications  of  Worm  Gearing.  —  Worm-wheels  are  used 
for  two  purposes.  They  may  be  employed  to  transmit  power 
where  it  is  desired  to  make  use  of  the  smoothness  of  action  which 
they  give,  and  the  great  reduction  in  velocity  of  which  they  are 
capable;  instances  of  this  application  of  worm  gearing  are  found 
in  the  spindle  drives  of  gear  cutters  and  other  machine  tools. 
They  are  also  used  where  a  great  increase  in  the  effective  power 
is  required;  in  this  case  advantage  is  generally  taken  of  the 


162  WORM  GEARING 

possibility  of  making  the  gearing  self-locking.  Such  service  is 
usually  intermittent  or  occasional,  and  the  matter  of  waste  of 
power  is  not  of  so  great  importance  as  in  the  first  case.  Exam- 
ples of  this  application  are  to  be  found  in  the  adjustments  of  a 
great  many  machine  tools,  in  training  and  elevating  gearing  for 
ordnance,  etc.  Calculations  for  the  general  design  of  this  class 
of  gearing  will  be  treated  separately  in  following  chapters.  In 
the  case  of  elevator  gearing  and  worm  feeds  for  machinery,  the 
functions  of  the  gearing  are,  in  a  measure,  a  combination  of  those 
in  the  two  applications. 

Examples  of  Worm  Gearing  Figured  from  the  Rules.  —  To 
show  how  the  rules  given  above  may  be  applied,  we  will  work 
out  two  examples.  The  first  of  these  is  for  a  light  machine  tool 
spindle  drive,  in  which  power  is  to  be  transmitted  continuously. 
It  is  determined  that  the  velocity  ratio  shall  be  8  to  i,  and  that 
the  proper  linear  pitch  to  give  the  strength  and  durability  re- 
quired shall  be  about  f  inch;  the  center  distance  is  required  to 
be  5  inches  exactly.  This  case  comes  under  the  first  of  the  two 
applications  just  described. 

Assume,  for  instance,  32  teeth  in  the  wheel,  and  a  quadruple- 
thread  worm.  We  will  figure  the  gearing  with  these  assumptions, 
and  see  if  it  appears  to  have  practical  dimensions. 

The  pitch  diameter  of  the  worm-wheel  by  Rule  10  is  found  to 
be 

=  7.6394  inches. 

The  pitch  diameter  of  the  worm  by  Rule  16  is  found  to  be 

(2  X  5)  —  7.6394  =  2.3606  inches. 
The  addendum  of  the  worm  thread  by  Rule  5  is  found  to  be 

0.3183  X  f  =  0.2387  inch. 
The  outside  diameter  of  the  worm  by  Rule  6  is  found  to  be 

2.3606  +  (2  X  0.2387)  =  2.8380  inches. 

For  transmission  gearing  the  angle  of  inclination  of  the  worm 
thread  should  not  be  less  than  18  degrees  or  thereabouts,  and 
the  nearer  30  or  even  40  degrees  it  is,  the  more  efficient  will  it  be. 
From  Rule  i  we  find  the  lead  to  be  4  X  1  =3  inches. 


RULES  AND  FORMULAS 


The  helix  angle  of  the  worm  thread  is  found  from  Rule  9  to 
be  2.3606  X  3-1416  -T-  3  =  2.4722  =  cot  22  degrees,  approxi- 
mately. This  angle  will  give  fairly  satisfactory  results.  The 
calculations  are  not  carried  any  further  with  this  problem, 
whose  other  dimensions  are  determined  from  those  just  found. 
In  the  following  case,  however,  all  the  calculations  are  made. 

For  a  second  problem  let  it  be  required  to  design  worm-feed 
gearing  for  a  machine  to  utilize  a  hob  already  in  stock.  This 

Dimensions  of  Worm-thread  Parts 


Number  of 
Threads  per 
Inch 

Circular  or 
Linear  Pitch, 
Inches 

Circ.  or  Lin. 
Pitch, 
Decimal 

Equivalents 

Height  of 
Tooth  above 
Pitch  Line 

Depth  of  Space 
below  Pitch 
Line 

Whole  Depth 
of  Tooth 

Thickness  of 
Tooth  on  Pitch 
Line 

Width  of 
Thread  Tool 
at  End 

Width  of 
Thread  at 
Top 

M 

2 

2  .  OOOO 

0.6366 

0.7366 

1-3732 

I  .  OOOO 

0.6200 

0.6708 

H 

l% 

1.7500 

0-5570 

0.6445 

1.2015 

0.8750 

0.5425 

0.5869 

% 

xM 

I  .  5000 

0-4775 

0.5524 

1.0299 

0.7500 

0.4650 

0.5031 

% 

iH 

1.2500 

0.3979 

0.4603 

0.8582 

0.6250 

0-3875 

0.4192 

i 

i 

I.  0000 

0.3183 

0.3683 

0.6866 

0.5000 

0.3100 

0-3354 

iH 

% 

0.7500 

0.2387 

0.2762 

0.5149 

0.3750 

0.2325 

0-2515 

IH 

% 

0.6667 

0.2122 

0.2455 

0-4577 

0-3333 

0.2066 

0.2236 

2 

W 

0.5000 

0.1592 

0.1841 

0-3433 

0.2500 

0.1550 

0.1677 

3H 

N 

0.4000 

0.1273 

0.1473 

0.2746 

O  .  2OOO 

o  .  i  240 

0.1341 

3 

H 

0.3333 

0.1061 

0.1228 

0.2289 

0.1667 

0.1033 

0.1118 

3M 

** 

0.2857 

o  .  0909 

0.1053 

0.1962 

0.1429 

0.0886 

0.0958 

4 

M 

0.2500 

0.0796 

0.0920 

0.1716 

0.1250 

0.0775 

0.0838 

4H 

% 

0.2222 

0.0707 

0.0819 

0.1526 

O.IIII 

0.0689 

0.0745 

5 

W 

O  .  2OOO 

0.0637 

0..0736 

o.i373 

O.IOOO 

0.0620 

0.0670 

6 

M 

0.1667 

0.0531 

0.0613 

0.1144 

0.0833 

0.0516 

0.0559 

7 

M 

0.1429 

0.0455 

0.0526 

0.0981 

0.0714 

0.0443 

0.0479 

8 

M 

0.1250 

0.0398 

o  .  0460 

0.0858 

0.0625 

0.0387 

0.0419 

9 

H 

O.IIII 

0.0354 

o  .  0409 

0.0763 

0.0556 

0.0344 

0.0373 

10 

Ho 

O.IOOO 

0.0318 

0.0369 

0.0687 

0.0500 

0.0310 

0.0335 

12 

Hi 

o  .  0833 

.  0.0265 

0.0307 

0.0572 

0.0416 

0.0258 

0.0279 

14 

H4 

0.0714 

0.0227 

0.0263 

0.0490 

0.0357 

O.O22I 

0.0239 

16 

He 

0.0625 

0.0199 

0.0230 

0.0429 

0.0312 

O.OI94 

0.0209 

18 

Ms 

0.0556 

0.0177 

0.0205 

0.0382 

0.0278 

O.OI72 

0.0186 

hob  is  double-threaded,  J  inch  linear  pitch,  and  2\  inches  diam- 
eter. The  center  distance  of  the  gearing  is  immaterial,  but  it  is 
decided  that  the  worm-wheel  ought  to  have  about  45  teeth  to 
bring  the  ratio  right.  The  only  calculations  made  are  those 
necessary  for  the  dimensions  which  would  appear  on  the  shop 
drawing. 
To  find  the  lead,  use  Rule  i:  0.5  X  2  =  i.o  inch. 


164 


WORM   GEARING 


To  find  the  whole  depth  of  the  worm  tooth,  use  Rule  3 :  0.5  X 
0.6866  =  0.3433  inch. 

To  find  the  addendum,  use  Rule  5:  0.5  X  0.3183  =  0.15915 
inch. 

To  find  the  pitch  diameter  of  the  worm,  use  Rule  7 :  2.5  —  2  X 
0.15915  =  2.1817  inches. 

To  find  the  bottom  diameter  of  the  worm,  use  Rule  8:  2.5  — 
2  X  0.3433  =  1.8134  inch. 

To  find  the  gashing  angle  of  the  worm-wheel,  use  Rule  9: 
2.18  X  3.14  -T-  i  =  6.845  =  c°t  &  degrees,  20  minutes,  about. 

To  find  the  pitch  diameter  of  the  worm-wheel,  use  Rule  10: 
45  X  0,5  -T-  3.1416  =  7.1620  inches. 


1-_1_ 


Machinery 


Fig.  4.     Shape  of  Blank  for  Worm 

To  find  the  throat  diameter  of  the  worm-wheel,  use  Rule  n: 
7.1620  +  2  X  0.15915  =  7.4803  inches. 

To  find  the  radius  of  the  throat  of  the  worm-wheel,  use  Rule  1 2 : 
(2.5  +  2)  -  (2  X  0.15915)  =  0.9317  inch. 

The  angle  of  face  may  be  arbitrarily  set  at,  say,  75  degrees,  in 
this  case.  The  "trimmed  diameter"  is  scaled  from  an  accu- 
rate drawing  and  proves  to  be  7.75  inches. 

To  find  the  distance  between  centers  of  the  worm  and  wheel, 
use  Rule  15:  (2.1817  +  7.1620)  -T-  2  =  4.6718  inches. 

To  find  the  minimum  length  of  threaded  portion  of  the  worm, 
use  Rule  17:  7.4803  -  4  X  0.15915  =  6.8437. 


V748032  —  6.843 72  =  3  inches,  approximately. 


RULES  AND  FORMULAS  165 

It  will  be  noted  that  the  ends  of  the  threads  in  Fig.  2  are 
trimmed  at  an  angle  instead  of  being  cut  square  down,  as  in  Fig.  i. 
This  gives  a  more  finished  look  to  the  worm.  It  is  easily  done  by 
applying  the  sides  of  the  thread  tool  to  the  blank  just  before 
threading,  or  it  may  be  done  as  a  separate  operation  in  preparing 
the  blank,  which  will  in  either  case  have  the  appearance  shown 
in  Fig.  4.  The  small  diameters  at  either  end  of  the  blank  in 
Fig.  4  should,  in  any  event,  be  turned  exactly  to  the  bottom 
diameter  shown  in  Fig.  2,  and  obtained  by  Rule  8.  This  is  of 
great  assistance  to  the  man  who  threads  the  worm,  as  he  knows 
that  the  threads  are  sized  properly  as  soon  as  he  has  cut  down  to 
this  diameter  with  the  end  of  his  thread  tool.  This  always 
requires,  of  course,  that  the  thread  tool  is  accurately  made. 

Formulas  for  the  Design  of  Worm  Gearing.  —  For  the  conven- 
ience of  those  who  prefer  to  have  their  rules  compressed  into 
formulas,  they  are  so  arranged  in  the  table  on  the  following 
pages.  The  reference  letters  used  are  as  follows: 

P  =  circular  pitch  of  wheel  and  linear  pitch  of  worm; 
/  =  lead  of  worm; 

n  =  number  of  teeth  or  threads  in  worm; 
S  =  addendum,  or  height  of  worm  tooth  above  pitch  line; 
d  =  pitch  diameter  of  worm; 
D  =  pitch  diameter  of  worm-wheel; 
o  =  outside  diameter  of  worm; 
0  =  throat  diameter  of  worm-wheel; 
O'  =  outside  diameter   of  worm-wheel  (to  sharp  corners) ; 
b  =  bottom  or  root  diameter  of  worm; 
N  =  number  of  teeth  in  worm-wheel; 
W  =  whole  depth  of  worm  tooth; 
T  =  width  of  thread  tool  at  end; 
a.  =  face  angle  of  worm-wheel; 
/3  =  helix  angle  of  worm  and  gashing  angle  of  wheel; 
U  =  radius  of  curvature  of  worm-wheel  throat; 
C  =  distance  between  centers; 
x  =  threaded  length  of  worm. 


i66 


WORM   GEARING 


Rules  and  Formulas  for  Worm  Gearing" 


To  Find 

Rule 

Formula 

Linear  Pitch. 

Divide  the  lead  by  the  num- 
ber of  threads.  —  It  is  under- 
stood that  by  the  number  of 
threads  is  meant,  not  number 
of  threads  per  inch,  but  the 
number  of  threads  in  the  whole 
worm  —  one,  if  it  is  single- 
threaded,  four,  if  it  is  quad- 
ruple-threaded, etc. 

p=l- 

n 

Addendum  of 
Worm  Tooth. 

Multiply  the  linear  pitch  by 
0.3183. 

5  =  0.3183  P 

Pitch  Diameter  of 
Worm. 

Subtract  twice  the  adden- 
dum from  the  outside  diam- 
eter. 

d  =  o-  2S 

Pitch    Diameter 
of  Worm-wheel. 

Multiply  the  number  of 
teeth  in  the  wheel  by  the  lin- 
ear pitch  of  the  worm,  and 
divide  the  product  by  3.1416. 

D        NP 

3.1416 

Center  Distance 
between  Worm  and 
Gear. 

Add  together  the  pitch  diam- 
eter of  the  worm  and  the  pitch 
diameter  of  the  worm-wheel, 
and  divide  the  sum  by  2. 

^_D  +  d 

2 

Whole    Depth    of 
Worm  Tooth. 

Multiply  the  linear  pitch  by 
0.6866. 

W=  0.6366  P 

Bottom  Diameter 
of  Worm. 

Subtract  twice  the  whole 
depth  of  tooth  from  the  out- 
side diameter. 

b  =  o  -  2  17 

Helix  Angle  of 
Worm. 

Multiply  the  pitch  diameter 
of  the  worm  by  3.1416,  and 
divide  the  product  by  the  lead; 
the  quotient  is  the  cotangent 
of  the  tooth  angle  of  the 
worm. 

rot/?_  3.i4i6  d 

i 

Width  of  Thread 
Tool  at  End. 

Multiply  the  linear  pitch  by 
0.31. 

r  =  o.3ip 

Throat  Diameter 
of  Worm-wheel. 

Add  twice  the  addendum  of 
the  worm  tooth  to  the  pitch 
diameter  of  the  worm-wheel. 

0  =  D  +  2S 

Radius  of  Worm- 
wheel  Throat. 

Subtract  twice  the  adden- 
dum of  the  worm  tooth  from 
half  the  outside  diameter  of 
the  worm. 

u  =  -2-*s 

From  MACHINERY'S  HANDBOOK. 


RULES  AND  FORMULAS 
Rules  and  Formulas  for  Worm  Gearing  —  (Continued} 


167 


To  Find 

Rule 

Formula 

Diameter  of 
Worm-wheel  to 
Sharp  Corners. 

Multiply  the  radius  of  curv- 
ature of  the  worm-wheel  throat 
by  the  cosine  of  half  the  face 
angle,  subtract  this  quantity 
from  the  radius  of  curvature, 
multiply  the  remainder  by  2, 
and  add  the  product  to  the 
throat  diameter  of  the  worm- 
wheel. 

o'  =  2(u-  ux 

cos-}  +0 
2/ 

Minimum  Length 
of  Worm  for  Com- 
plete Action. 

Subtract  four  times  the  ad- 
dendum of  the  worm  thread 
from  the  throat  diameter  of 
the  wheel,  square  the  remain- 
der, and  subtract  the  result 
from  the  square  of  the  throat 
diameter  of  the  wheel.  The 
square  root  of  the  result  is  the 
minimum  length  of  worm  ad- 
visable. 

x  =  VO2-(O-45)2 

Outside  Diameter 
of  Worm. 

Add  together  the  pitch  diam- 
eter and  twice  the  addendum. 

0  =  J+25* 

Pitch    Diameter 
of  Worm. 

Subtract  the  pitch  diameter 
of  the  worm-wheel  from  twice 
the  center  distance. 

d  =  2C  -D 

Worms  with  Large  Helix  Angle.  —  When  worms  have  a  large 
helix  angle  (15  degrees  or  more)  the  dimensions  of  the  thread 
should  be  measured  at  right  angles  to  the  helix.  In  such^cases, 
the  following  changes  should  be  made  in  the  formulas  in  the 
table  and  in  the  corresponding  rules.  Let: 

Pn  =  normal  circular  pitch  =  P  X  cos  0. 

Then  the  formulas  giving  addendum,  whole  depth  of  worm 
tooth  and  width  of  thread  tool  at  end  will  be  written  as  follows : 
S  =  0.3183  Pn;        W  =  0.6866  Pn;        r  =  o.3iPn. 

When  these  changes  are  made  all  the  other  formulas  will  give 
correct  results  when  used  in  their  original  form. 

Table  for  Calculating  the  Outside  Diameter  of  Worm-wheels. 
—  The  regular  formula  for  calculating  the  outside  diameter  (to 
sharp  corners)  of  a  worm-wheel  is: 

0'  = 


i68 


WORM  GEARING 


in  which  Of  =  the  outside  diameter  of  worm-wheel  to  sharp 
corners;  U  =  the  radius  of  the  curvature  of  worm- wheel  throat; 
a  =  face  angle  of  worm-wheel;  O  =  throat  diameter  of  worm- 
wheel. 

By  writing  this  formula  in  the  form: 


it  will  be  seen  that  the  expression  within  the  parentheses  can  be 
tabulated  for  various  face  angles,  and  such  a  table  is  given  here- 
Table  of  Factors  C  Used  in  Worm-gear  Formula 


-0 

\  riP 

jjR 

-jj 

||> 

ih  — 

— 

-o  

—  HJ 

K 

Angle  a, 

Factor 

Angle  a, 

Factor 

Angle  a, 

Factor 

Angle  a, 

Factor 

Degrees 

C 

Degrees 

C 

Degrees 

C 

Degrees 

C 

30 

0.034 

46 

0.080 

62 

0.143 

78 

0.223 

31 

0.036 

47 

0.083 

63 

0.147 

79 

0.228 

32 

0.039 

48 

0.086 

64 

0.152 

80 

0.234 

33 

0.041 

49 

0.090 

65 

0.157 

81 

o.  240 

34 

0.044 

50 

0.094 

66 

0.161 

82 

0.245 

35 

0.046 

0.097 

67 

0.166 

83 

0.251 

36 

0.049 

52 

O.  101 

68 

0.171 

84 

0.257 

37. 

0.052 

53 

0.105 

69 

0.176 

85 

0.263 

38 

0.054 

54 

0.109 

70 

0.181 

86 

o.  269 

39 

0.057 

55 

0.113 

71 

0.186 

87 

0.275 

40 

0.060 

56 

0.117 

72 

0.191 

88 

0.281 

0.063 

57 

O.I2I 

73 

0.196 

89 

0.287 

42 

0.066 

58 

0.125 

74 

O.2OI 

90 

0.293 

A'J 

O   O7O 

^O 

o  .  130 

75 

o.  207 

AA 

V-*  .  W/  W 

00*77 

60 

o  .  1  34 

/  o 
76 

0  .  21  2 

•Wo 

f" 

A  £ 

o  .  076 

61 

o  .  138 

77 

O    217 

/  / 

vy  •  *  A  / 

with.     By  using  this  table  and  calling  the  values  found  in  the 
table  for  various  angles  C,  the  formula  takes  the  simple  form: 

Of  =  2  U  X  C  +  O, 

in  which  C  can  be  found  in  the  table  for  any  angle  from  30  to  90 
degrees. 

Model  Worm-gear  Drawing.  —  A  model  drawing  of  a  worm- 
wheel  and  worm,  properly  dimensioned,  is  shown  in  Fig.   5. 


RULES  AND  FORMULAS 


169 


This  drawing  follows,  in  general,  the  model  drawings  shown  by 
Mr.  Burlingame  in  the  August,  1906,  issue  of  MACHINERY, 
taken  from  the  drafting-room  practice  of  the  Brown  &  Sharpe 
Mfg.  Co.  In  cases  where  the  worm-wheel  is  to  be  gashed  on  the 
milling  machine  before  bobbing,  the  angle  at  which  the  cutter 
is  set  should  also  be  given.  This  is  the  same  as  the  angle  of  worm 


h i%" — H 


-3ya" J 


„_}_ 


WHEEL 

NUMBER  OF  TEETH  =45 

CIRCULAR  PITCH=*0.500" 

ANGLE  OF  CVT=8°2Qf 

WORM,  DOUBLE,  R.  H. 

OUTSIDE  DIAM.  OF  WORM  =  2.500" 


Machinery 


Fig.  5.    Model  Drawing  of  Worm  and  Worm-wheel 

tooth  found  by  Rule  9.  In  cases  where  the  wheel  is  to  be  hobbed 
directly  from  the  solid  by  a  positively  geared  nobbing  machine, 
this  information  is  not  needed.  It  might  be  added  that  it  is 
impracticable  with  worm-wheels  having  less  than  16  or  18  teeth 
to  gash  the  wheel,  and  then  hob  it  when  running  freely  on  centers, 
if  the  throat  diameter  has  been  determined  by  Rule  n. 


CHAPTER  VII 
ALLOWABLE  LOAD  AND  EFFICIENCY  OF  WORM  GEARING 

WHEN  called  upon  to  design  a  set  of  worm  gearing  for  a  cer- 
tain drive,  or  select  one  from  the  catalogue  of  a  manufacturer, 
the  designer  will  find  very  little  definite  information  in  the 
ordinary  textbooks  on  machine  design  concerning  the  allowable 
load,  the  allowable  speed  and  the  efficiency  which  may  be  ex- 
pected—  the  very  points  which  are  of  vital  interest  to  him. 
The  following  paragraphs  discuss  these  subjects. 

Relation  of  Load  to  Effort  —  In  the  following  formulas  let 
P  =  pressure  of  the  worm-wheel  on  the  worm  parallel  to 

the  worm-shaft; 

F  =  force  which  must  be  applied  at  the  pitch  radius  of  the 
worm  at  right  angles  to  the  worm-shaft  to  overcome 

P\ 
a.  —  angle  of  thread  with  a  line  at  right  angles  to  the  axis 

of  the  worm; 
/  =  coefficient  of  friction; 
/  =  lead  of  worm  thread; 
d  =  pitch  diameter  of  worm. 

The  normal  pressure  between  worm  and  wheel  then  equals 
F  X  sin  a  +  P  X  cos  a,  and  the  friction  /  (F  X  sin  a  +  P  X 
COS  a). 

Now,  if  the  worm  is  revolved  once,  we  obtain  the  following 
relation  between  F  and  P: 

FXird  =  PI  +f(F  X  sin  a  +  P  X  cos  a) 


cos  a 

As  —  :  =  tan  a.  the  formula  above  may  be  written: 
ird 


170 


LOAD  AND  EFFICIENCY 


171 


This  relation,  giving  the  force  F  which  must  be  applied  at  the 
pitch  radius  of  the  worm  to  overcome  the  load  P  at  the  pitch 
radius  of  the  worm-gear,  is  often  required  by  the  designer. 

Efficiency.  —  If  there  were  no  friction,  or  if  /  equalled  0,  we 
would  have  : 

Fi  =  P  tan  a. 


f  , 


The  efficiency  of  the  worm  gearing,  is,  therefore: 

,.,      FI      tan«(i  -/tana) 
£=F=         /+tan« 

Equations  (i)  and  (2)  are,  strictly  speaking,  only  correct 
for  worm  threads  with  vertical  sides,  but  the  sloping  thread  side 
commonly  used  affects  the  result  but  little. 


10 


20  30  40  50  GO  70 

ANGLE  OF  WORM  THREAD,DEGREES 


80  90 

MacMnery,N.Y. 


Fig.  i.    Relation  between  Worm  Thread  Angle  and  Efficiency 

To  demonstrate  the  influence  of  the  thread  angle  on  the 
efficiency,  the  curve  represented  by  Equation  (2)  with  a  and  E 
as  variables,  and  for  a  certain  assumed  value  of  /,  has  been 
plotted  in  Fig.  i.  This  curve  is  reproduced  from  "Worm  and 
Spiral  Gearing,"  by  F.  A.  Halsey.  It  shows  that  the  efficiency 
increases  very  rapidly  with  the  thread  angle  for  small  angles, 
while  for  angles  near  the  maximum  efficiency,  there  is  very  little 
drop  for  a  wide  range  of  angles.  It  is,  therefore,  essential,  for 
high  efficiency,  not  to  use  thread  angles  that  are  too  small.  The 
value  of/  is  found  by  experiments  to  vary  with  the  speed  of  the 
rubbing  surfaces.  (See  Transactions  of  the  American  Society 


172 


WORM  GEARING 


of  Mechanical  Engineers,  Vol.  7,  page  273.)  For  values  of  / 
for  various  speeds  see  table,  "Safe  Load  on  Worm-gear  Teeth." 
Allowable  Load.  —  It  would  seem  reasonable  to  assume  the 
allowable  pressure  on  gear  teeth  under  otherwise  equal  conditions 
to  be  expressed  by 

(3) 


where  p  =  pitch,  b  =  width  of  gear  teeth,  and  C  =  a  constant 
for  the  given  speed. 

For  very  slow  speed,  where  there  is  no  danger  of  overheating, 
and  where  the  only  questions  to  be  taken  into  account  are  the 
strength  and  the  resistance  to  abrasion,  the  above  equation  is 

Relation  Between  Velocity  at  Pitch  Line,  Angle  of  Thread  and  Efficiency 


Angle  of  Thread,  Degrees 

Velocity  at  Pitch 
Line,  Feet  per 

5 

10 

20 

30 

40 

45 

Minute  • 

Efficiency,  Per  Cent 

5 

40 

56 

69 

76 

79 

80 

10 

47 

62 

74 

79 

82 

82 

20 

52 

67 

78 

83 

85 

86 

30 

56 

7i 

81 

85 

87 

87 

40 

60 

74 

83 

87 

88 

88 

So 

63 

76 

85 

88 

89 

89 

75 

67 

80 

87 

90 

90 

90 

100 

70 

82 

88 

9i 

9i 

9i 

150 

74 

84 

90 

92 

92 

92 

200 

76 

85 

Qi 

92 

92 

92 

obviously  correct  if  we  assume  some  standard  ratio  of  worm 
diameter  to  wheel  face.  A  pitch  diameter  of  worm  equal  to  1.5 
times  the  face  of  the  wheel  (corresponding  to  a  face  angle  of  76 
degrees)  is  about  right. 

For  higher  speeds  the  amount  of  frictional  work  transformed 
into  heat  may  cause  an  excessive  rise  in  temperature  before  the 
worm,  or  casing  surrounding  the  worm,  is  able  to  carry  off  the 
same  amount  of  heat  as  is  developed,  and  the  limiting  load  will 
be  determined  thereby.  The  ability  of  the. worm  or  casing  to 
carry  off  heat  is  proportional  to  the  surface,  which,  again,  is 
approximately  proportional  to  the  product  pb  of  the  pitch  and 
width  of  the  gear  teeth;  hence,  Equation  (3)  is  approximately 


LOAD  AND  EFFICIENCY 


173 


correct  in  this  case  also.      This  equation  is  given  in  "Des  Ingen- 
ieurs  Taschenbuch"  (Hutte). 

German  Experiments  to  Determine  Speed  Factor.  —  The 
value  of  the  factor  C  varies  with  the  speed  and  must  be  deter- 
mined by  experiment.  The  most  complete  experiments  to  this 
effect  are  those  of  C.  Bach  and  E.  Roser,  published  in  Zeit- 
schrift  des  Vereines  Deutscher  Ingenieure,  Feb.  14,  1903.  These 
experiments  were  made  with  a  three-threaded  steel  worm,  not 
hardened;  76.6  millimeters  (3  inches)  pitch  diameter;  25.4  milli- 
meters (i  inch)  pitch;  17  degrees  34  minutes,  thread  angle; 
148  millimeters  (5^!  inches)  long.  The  worm-wheel  was  of 
bronze,  242.6  millimeters  (g^g  inches)  pitch  diameter,  with 


'0.260.780.41  2.77  5.40 

SLIDING  VELOCITY  IN  METERS  PER  SECOND 
MEASURED  AT  THE  PITCH  CIRCLE  OF  THE  WORM 


70°  C. 


50   C. 


8.61  METERS 


Machinery  ,N.  Y. 


Fig.  2.    Relation  between  Tangential  Pressure  and  Velocity 


milled  teeth,  78  millimeters  ($•£$  inches)  wide  measured  on  the 
arc,  30  teeth,  speed  ratio  i  to  10,  with  ball  bearings  for  worm 
shaft,  and  oil  bath  of  extremely  viscous  oil. 

In  the  experiments  the  load  on  the  teeth  varied  from  in  kil- 
ograms (244  pounds)  to  1257  kilograms  (2765  pounds)  and  the 
speed  varied  from  2185  R.P.M.  to  64  R.P.M.  The  temperature 
of  the  oil  bath  and  that  of  the  surrounding  air  was  observed  until 
the  difference  reached  a  constant  value.  Corresponding  values 
of  load  and  speed  for  constant  temperature  difference  were  ascer- 


174 


WORM   GEARING 


tained  and  an  attempt  to  express  the  relation  by  an  equation 
gave  the  following  rather  lengthy  expression: 

P  =  Cpb  =  [a(ta-te)+d]pb  (4) 

in  which 

o.o66c 
V 


a  = 


+  0.4192; 


d 


109.1 

+  2-75 


-  24.92; 


t0  =  temperature  of  oil  in  degrees  C. ; 
te  =  temperature  of  air  in  degrees  C. ; 
V  =  sliding  velocity  at  pitch  line  in  meters  per  second. 

Safe  Load  on  Worm-gear  Teeth 

Load  per  unit  of  the  product  (pitch  X  width  of  tooth),  for  go-degree  F. 
temperature  difference  between  oil  and  surrounding  air.  More  than  1000 
pounds  per  unit  of  product  (pitch  X  width  of  tooth)  should  not  be  allowed 
under  ordinary  circumstances.  Cut  bronze-gear,  cut  steel-worm. 


Velocity  in 
Feet  per 
Minute 

Load  in 
Pounds  per 
Unit  of 
(Pitch  X 
Width  of 
Tooth) 

Coefficient 
of  Friction 

Velocity  in 
Feet  per 
Minute 

Load  in 
Pounds  per 
Unit  of 
(Pitch  X 
Width  of 
Tooth) 

Coefficient 
of  Friction 

5 

IOOO 

0.146 

2OO 

403 

IO 

IOOO 

o.  116 

271 

20 

0^6 

'    o  .  ooo 

3OO 

241 

7QO 

•^o 

40 

5° 

703 
646 

0.070 

400 
SOO 

292 

2^7 

75 

564 

600 

228 

80 

554 

0.054 

700 

206 

IOO 

800 

iBc 

12"? 

477 

ooo 

167 

175 

448 

424 

IOOO 
I2OO 

128 

The  curves  represented  by  Equation  (4)  are  distinctly  hyper- 
bolic in  character.  Two  of  these  curves  have  been  plotted  in 
Fig.  2,  one  for  a  temperature  difference  of  50  degrees  C.  and  one 
for  70  degrees  C. 

The  accompanying  table  shows  the  loads  for  various  speeds, 
calculated  from  Equation  (4)  and  transformed  into  English 
units,  for  a  difference  in  temperature  of  90  degrees  F.  (50  de- 
grees C.).  In  the  same  table  are  given  coefficients  of  friction  as 
deduced  by  Unwin  from  Lewis's  experiments.  This  table  of 


LOAD  AND  EFFICIENCY  175 

load  used  with  discretion,  and  with  due  consideration  for  the 
various  individual  conditions  associated  with  the  drive  in  con- 
templation, may  be  made  the  basis  for  worm-gear  design  in 
average  cases  and  where  a  temperature  rise  of  90  degrees  F. 
(50  degrees  C.)  is  allowable.  The  loads  given  are  for  continu- 
ous service,  and  as  it  will  take  several  minutes,  perhaps  hours, 
before  the  constant  temperature  is  reached,  a  higher  load  will 
be  justified  for  intermittent  service,  where  the  oil  has  time  to 
cool  down.  It  should  be  kept  in  mind  that  the  danger  of 
abrasion  will,  of  course,  depend  on  the  temperature  of  the  oil, 
and  not  on  the  temperature  difference;  if,  therefore,  the  gear- 
ing is  installed  in  a  place  where  the  surrounding  temperature 
is  kept  low,  the  temperature  difference  can  be  correspondingly 
increased  and  vice  versa.  The  danger  of  abrasion  will  also,  to 
a  large  extent,  depend  on  the  character  of  the  lubricant,  in  that 
a  very  viscous  oil  will  offer  greater  resistance  to  the  squeezing 
out  of  the  oil  film  between  the  rubbing  surfaces  than  the  less 
viscous. 

Practical  Points  in  the  Design  of  Worm  and  Gear.  —  It  should 
be  remembered  also  that  a  gear  with  many  teeth  gives  a  better 
contact  with  the  worm  both  on  account  of  the  flatter  curve  of 
the  engaging  segment  and  the  larger  average  radii  of  curvature 
of  its  teeth.  This  has  particular  reference  to  the  heavy  loads  at 
slow  speed,  where  the  question  of  temperature  does  not  enter. 

The  angle  of  thread  (the  helix  angle)  does  not  appear  in  the 
formula  given,  as  it  has  no  direct  bearing  on  the  question  of 
allowable  load  and  speed  of  rubbing  surfaces.  As  previously 
mentioned,  the  angle  of  thread  has,  however,  a  direct  influence 
on  the  efficiency  of  the  gearing.  Given,  for  instance,  two  worms 
of  the  same  diameter,  one  having  a  thread  angle  twice  as  great 
as  the  other,  carrying  the  same  load  on  the  gear  teeth  and  run- 
ning at  the  same  speed,  there  is  no  reason  at  all  why  one  should 
be  more  successful  than  the  other  as  far  as  wearing  qualities 
are  concerned,  but  it  must  be  remembered  that  the  first  one  is 
transmitting  twice  the  horsepower  of  the  other,  and  will  obvi- 
ously give  much  better  efficiency. 

With  the  allowable  load  decreasing  as  the  speed  increases,  as 


1  76  WORM  GEARING 

provided  for  by  the  formula  and  table  given,  a  speed  of  rubbing 
surfaces  as  high  as  1000  feet  per  minute,  or  even  higher,  can 
undoubtedly  be  used  with  success  for  cut  gearing,  which  also  has 
been  demonstrated  repeatedly  in  practice.  In  the  tests  by 
Bach  and  Roser,  the  speed  was  carried  as  high  as  8.76  meters 
per  second  (1724  feet  per  minute),  with  a  load  of  370  kilograms 
(814  pounds)  and  a  temperature  difference  of  80.5  degrees  C. 
(126.9  degrees  F.)  with  no  apparent  cutting.  The  loads  given 
represent  tangential  loads  at  right  angles  to  the  worm-gear  shaft. 
The  actual  pressure  between  the  rubbing  surfaces  will  be  more,  and 
will  increase  with  the  angle  of  thread,  but  the  increase  for  gears 
in  common  use  (less  than  2o-degree  thread  angle)  is  not  very  great. 

Concerning  the  coefficient  of  friction  /,  this  has  not  been  de- 
duced for  higher  speeds  than  80  feet  per  minute,  but  it  will  be 
seen  that  there  is  a  general  tendency  for  the  value  of/  to  decrease 
as  the  speed  increases. 

Except  for  hand-operated  gearing,  or  for  machinery  which  is 
only  operated  occasionally  and  for  a  very  short  time,  the  worm 
and  gear  should  be  enclosed  in  an  oil  casing  and  the  worm  always 
placed  below  the  gear  to  insure  the  submersion  of  the  rubbing 
surfaces  in  oil.  Except  in  the  cases  mentioned,  machine-cut 
worms  and  wheels  should  always  be  used.  Hardened  steel 
worms  working  with  bronze  wheels  have  proved  to  give  good 
satisfaction,  because  this  combination  wears  longer  than  cast 
iron  or  steel  and  cast  iron. 

Self-locking  Worm  Gearing.  —  A  set  of  worm  gearing  will  be 
self-locking  when  the  thread  angle  is  equal  to,  or  smaller  than, 
the  angle  of  friction.  From  Equation  (2)  we  obtain,  by  making 
/  =  tan  a,  the  efficiency  of  worm  gearing  having  a  thread  angle 
just  small  enough  to  be  self  -locking,  as  follows: 

tan  a  (i  —  tan2  a        -,  ,  ,  * 

(5) 


Equation  (5)  gives  a  maximum  of  EI  for  tan  a  =  o  or  a  =  o, 
and  this  value  is  E\  max.  =  \. 

From  this  it  will  be  seen  that  it  is  impossible  to  obtain  an 
efficiency  greater  than  0.5  if  the  gears  are  to  be  self  -locking  in 


LOAD  AND  EFFICIENCY  177 

themselves.  Of  course,  there  will  always  be  some  friction  in 
the  worm-shaft  bearings  and  other  parts  of  the  machinery  which 
may  prevent  the  pressure  on  the  worm-gear  from  actually  turn- 
ing the  machinery  as  a  whole  backwards,  even  if  the  angle  of 
thread  is  larger  than  that  of  friction.  This,  in  connection  with 
the  fact  that  the  efficiency  for  backward  movement  is  low,  is 
probably  the  reason  why  many  worm-gear  drives,  applied  as 
self-locking,  have  angles  of  thread  far  in  excess  of  the  friction 
angle,  and  still  seem  to  work  satisfactorily. 

On  account  of  the  variable  coefficient  of  friction,  the  angle  of 
thread  which  may  safely  be  used  for  self-locking  gears  will  also 
depend  largely  on  the  speed  with  which  the  machine  is  run  back- 
ward, or,  in  other  words,  the  speed  with  which  the  load  is  lowered 
or  eased  off  by  means  of  the  worm-gear.  If  the  machine  is  never 
run  but  one  way,  and  the  worm-gear  applied  as  safety  device  to 
prevent  backward  movement  in  case  of  accident,  then  the  load 
would  have  to  start  the  worm  shaft  rotating,  and  a  larger  angle  of 
thread  could  undoubtedly  be  used.  The  subject  of  self-locking 
worm  gearing  will  be  treated  in  greater  detail  in  a  following  chapter. 

An  Example  from  Practice.  —  To  indicate  the  use  of  the  form- 
ulas and  table  in  practical  work,  the  following  example  has  been 
prepared:  Assume  a  set  of  worm  gearing  used  for  driving  a 
package  elevator  with  the  worm-gear  shaft  running  at  a  speed 
of  5  R.P.M.  The  required  turning  moment  is  42,000  inch- 
pounds.  It  is  desired  to  have  the  worm  gearing  self-locking  to 
prevent  the  elevator  from  running  backward  in  case  the  driving 
belt  breaks  or  jumps  off. 

As  the  elevator  must  come  to  a  stop  before  it  can  commence 
to  run  backward  it  is  only  necessary  to  have  a  thread  angle  equal 
to  or  smaller  than  the  angle  of  friction  for  rest.  Assuming  the 
coefficient  of  friction  to  be  at  least  0.15  at  rest,  the  thread  angle 
a  will  be  determined  by  tana  =  0.15,  or  a.  =  8|  degrees.  If 
the  speed  of  the  worm  shaft  is  not  dependent  on  other  conditions, 
we  have  a  choice  between  a  single-  and  a  double-threaded  worm. 
A  single-threaded  worm  of  if  inch  pitch  would  have  a  diameter 

=       *     =  3.71  inches.    This  may  not  be  enough  to  allow  for 


178  WORM  GEARING 

a  worm  shaft  of  sufficient  strength;  besides  it  would  give  a  very 
narrow  face  to  the  worm-gear.  We,  therefore,  probably  prefer 
to  use  a  double -threaded  worm,  the  pitch  diameter  of  which 

i-  X  2 
will  be  — =  7.42  inches.    The  face  of  the  worm-gear  will 

then  be 

f  X  7.42  =  4-95  inches,  or,  say,  5  inches. 

Assuming  a  worm-gear  of  28  inches  pitch  diameter,  if  inch 
pitch,  and  50  teeth,  the  worm  shaft  will  be  running  5  X  -/-  = 

125  R.P.M.,  which  gives  a  speed  of  rubbing  of n-.   , — * 

12  X  cos8f  deg. 

=  247  feet  per  minute. 

Referring  to  the  table,  "Safe  Load  on  Worm-gear  Teeth," 
we  find  for  a  speed  of  250  feet  per  minute  an  allowable  load  of 
371  pounds  per  unit  of  product  (pitch  X  width  of  tooth).  The 
total  allowable  load  in  this  case  will  be37iXi|X5=  3246 
pounds.  This  load  at  14  inches  radius  gives  a  turning  moment 
of  14  X  3246  =  45,444  inch-pounds,  while  only  42,000  inch- 
pounds  is  required. 

If  the  above  machine  were  applied  for  lowering  packages  in- 
stead of  elevating  same,  as  previously  assumed,  the  gearing  would 
have  to  lock  while  running  at  a  full  speed  of  247  feet  per  minute, 
at  which  speed  we  would  not  have  a  friction  coefficient  of  more 
than  0.05,  at  the  most,  which  would  correspond  to  an  angle  of 
thread  determined  by  tan  /3  =  0.05,  or  0  =  2  degrees  50  minutes 
approximately,  and  the  gear  with  a  thread  angle  of  8|  degrees 
could  not  be  expected  to  lock. 

To  find  the  efficiency  of  the  above  gearing  when  running  at 
full  speed,  assume  a  coefficient  of  friction  of  0.05,  and  apply 
Formula  (2)  which  gives 

„      tana  (i —/tana)      0.15(1—0.05X0.15) 

E  =  -    — ^ — J—    —  =  — •" -^  =  74  per  cent. 

/  +  tana  0.05  +  0.15 

This  is  the  efficiency  of  the  worm  gearing  only  and  does  not 
allow  for  the  friction  loss  in  the  worm-gear  shaft  nor  any  fric- 
tional  loss  in  the  other  parts  of  the  machine. 

To  find  the  effort  F  which  must  be  exerted  at  the  pitch  radius 


LOAD  AND  EFFICIENCY 


I79 


of  the  worm  to  turn  the  worm  shaft  with  a  load  = 


OOO 

-  - 
14 


=  3000 


pounds,  at  the  worm-gear  periphery,  apply  Formula  (i)  which 
gives  (for/  =  0.15  at  starting): 


P=  P  X 


=  3000  X 


921  pounds. 


i  —/tana  1—0.15X0.15 

To  this  should  be  added  the  friction  in  the  worm-shaft  bearings 
reduced  to  the  same  radius. 

Theoretical  Efficiency  of  Worm  Gearing  —  Oerlikon  Experi- 
ments. —  The  following  table  gives  the  theoretical  efficiency  of 
worm  gearing  for  a  number  of  different  coefficients  of  friction. 
Practical  experiments  carried  out  by  the  Oerlikon  Company, 
Oerlikon  by  Zurich,  Switzerland,  agree  closely  with  the  results 
from  theoretical  calculations  given  in  the  table.  These  experi- 
ments indicate  that  the  efficiency  increases  with  the  angle  of 
Table  Giving  Theoretical  Efficiency  of  Worm  Gearing 


Coefficient  of 
Friction 

Angle  of  Inclination 

Sdeg. 

10  deg. 

IS  deg. 

20  deg. 

25  deg. 

30  deg. 

35  deg. 

40  deg. 

45  deg. 

O.OI 

8Q.7 

94-5 

96.1 

97.0 

97-4 

97-7 

97-9 

98.0 

98.0 

O.O2 

8l-3 

89-5 

92.6 

94-1 

95-o 

95-5 

95-9 

96.0 

96.  1 

0.03 

74-3 

85.0 

8Q.2 

91.4 

92.7 

93-4 

93-9 

94-1 

94.2 

O.O4 

68.4 

80.9 

86.1 

88.8 

90.4 

91.4 

92.0 

92.2 

92-3 

O.OS 

63-4 

77-2 

83-1 

86.3 

88.2 

89.4 

90.1 

90.4 

90-5 

O.O6 

S9-o 

73-8 

80.4 

84.0 

86.1 

87-.S 

88.2 

88.6 

88.7 

O.O7 

55-2 

70.7 

77-8 

81.7 

84.1 

85-6 

86.4 

86.9 

86.9 

0.08 

Si-9 

67.8 

75-4 

79.6 

82.2 

83.8 

84.7 

85.2 

85-2 

O.OQ 

48.9 

65.2 

73-i 

77-6 

80.3 

82.0 

83.0 

83.5 

83.5 

O.IO 

46.3 

62.7 

70.9 

75-6 

78-5 

80.3 

81.4 

81.9 

81.8 

inclination,  up  to  a  certain  point.  They  also  show  that  for 
larger  angles  of  inclination  than  from  25  to  30  degrees  the  effi- 
ciency increases  very  little,  especially  if  the  coefficient  of  fric- 
tion is  small,  and  this  fact  is  of  importance  in  practice,  because, 
for  reasons  of  gear  ratio  and  conditions  of  a  constructive  nature, 
an  angle  greater  than  30  degrees  cannot  be  employed.  The 
coefficient  of  friction  increases  with  the  load  and  diminishes  to 
a  certain  extent  with  the  increase  of  speed.  Besides  the  friction 
between  the  worm  and  the  wheel  teeth,  there  is  also  the  friction 
of  the  spindle  bearings  and  the  ball  bearings  for  taking  the  axial 


l8o  WORM   GEARING 

thrust.  To  obtain  the  best  results,  there  must  be  very  careful 
choice  of  dimensions  of  teeth,  of  the  stress  between  them,  and 
the  angle  of  inclination.  To  show  what  can  be  done,  the  follow- 
ing are  the  results  of  a  test  with  an  Oerlikon  worm-gear  for  a 
colliery  winding  engine:  The  motor  gave  30  brake  horsepower 
to  40  brake  horsepower  at  780  revolutions.  The  normal  load 
was  25  brake  horsepower,  but  at  starting  it  could  develop  40 
brake  horsepower.  The  worm-gear  ratio  was  13.6  to  i,  the 
helicoidal  bronze  wheel  having  68  teeth  on  a  pitch  circle  of  7.283 
inches,  and  the  worm  5  threads.  The  power  required  at  no  load 
for  the  whole  mechanism  was  520  watts,  corresponding  to  2.8 
per  cent  of  the  normal.  The  efficiency  at  one-third  normal  load 
gave  90  per  cent,  at  full  load  94^,  and  at  50  per  cent  overload 
93  per  cent.  The  efficiency  of  the  worm  and  wheel  alone  is  higher, 
and,  knowing  the  no-load  power,  is  calculated  to  be  97!  per  cent. 
According  to  the  table  given,  of  theoretical  efficiencies,  this  gives 
the  coefficient  of  friction  as  o.oi.  To  obtain  a  reduction  of  13.6 
to  i  with  spur  gears  would  have  necessitated  two  pinions  and 
two  wheels  with  their  spindles  and  bearings,  and  if  the  bearing 
friction  was  taken  into  consideration,  the  efficiency  of  such  gear- 
ing would  certainly  not  have  reached  the  above-mentioned  figure 
of  94^  per  cent  at  full  load.  These  figures,  of  course,  seem  very 
high  for  the  efficiency  of  worm  gearing.  They  were  published 
in  MACHINERY,  December,  1903,  having  been  obtained  from  a 
reliable  source,  and  were  never  challenged.  They  have  also  been 
published  in  several  editions  of  MACHINERY'S  Reference  Book 
No.  i,  "Worm  Gearing,"  without  adverse  criticism. 

Worm  and  Helical  Gears  as  Applied  to  Automobile  Rear- 
axle  Drives.  —  European  practice  extending  over  a  period  of 
fifteen  years  has  given  ample  evidence  of  the  eminent  success  of 
the  worm  and  helical  type  of  gearing,  and  in  a  paper  read  before 
the  Society  of  Automobile  Engineers,  Mr.  F.  Burgess,  the  well- 
known  gear  expert,  stated  that  he  felt  confident  in  saying  that 
in  the  near  future  a  large  percentage  of  the  cars  in  the  United 
States  will  be  equipped  with  this  drive.  The  principal  reason 
for  the  adoption  of  the  helical  form  of  tooth  appears  to  be  its 
peculiar  quality  of  silence,  regardless  of  speed  or  load.  With 


LOAD  AND  EFFICIENCY 


181 


the  best  methods  of  design  and  assembly,  great  durability, 
strength  and  efficiency  are  obtained. 

The  successful  worm-gear  should  embody  the  following  quali- 
fications : 

1.  Cheapness  of  construction. 

2.  Strength  for  resisting  shocks. 

3.  Hardened  and  smooth  surfaces  for  durability. 

4.  Material  of  a  suitable  composition  to  reduce  friction. 

5.  Simplicity  of  construction  and  mounting. 

6.  Perfect  bearing  conditions. 

7.  Noiselessness  at  any  speed  or  load. 

8.  Reversibility. 

9.  Lightness  in  weight. 

10.   High  efficiency  in  power  transmission. 

Granting  that  there  is  some  argument  against  the  worm  in 
regard  to  trucks  as  to  the  dead  axle  proposition,  this  could  be 
overcome  by  using  a  worm-gear  on  each  end  of  the  axle,  the 

Results  of  Efficiency  Tests  on  Ordinary  Type  Worm-gear  for  Auto- 
mobile Rear-axle  Drive  for  Electric  Vehicles  and  Light- 
power  Cars 


Temper- 

Input, 

Number 
of 
Test 

ature  of 
Worm- 
gear, 
Degrees 

Twist  of 
Shaft, 
Degrees 

R.P.M. 
of 
Worm 

R.P.M. 
of 

Worm- 
gear 

Transmis- 
sion Dyna- 
mometer, 
Horse- 

Output, 
Brake 
Horse- 
power 

Effi- 
ciency, 
Per 
Cent 

F. 

power 

I 

74 

iH 

1393 

143 

1  .64 

I  .Ol 

6l.6 

2 

82 

2 

1423 

146 

2.65 

2.  II 

7Q.6 

3 

86 

2% 

1416 

145 

3-41 

3-H 

91-3 

4 

86 

39/ie 

1416 

145 

4.46 

4.15 

93 

5 

QO 

4^6 

1370 

140.5 

5.48 

5-03 

92 

6 

94 

57/ie 

1389 

142.5 

6.72 

6.12 

91.2 

Worm-gear:  Phosphor-bronze,  39  teeth. 

Worm:  Casehardened  steel  worm,  solid  on  shaft,  quadruple  thread. 

same  as  sprocket  wheels,  having  a  double  worm-gear  drive  in 
place  of  the  cumbersome  chain  drive.  If  at  first  this  is  slightly 
more  expensive  than  the  chain  and  sprocket  drive,  less  repairs 
will  more  than  make  up  the  difference.  Care  should  be  taken 
to  have  accurate  bearings,  both  radial  and  end-thrust. 


182 


WORM   GEARING 


Considerable  discussion  has  arisen  in  regard  to  the  relative 
merits  of  the  straight  and  Hindley  types  of  worm  gearing.  Both 
can  be  used  successfully,  although  each  has  its  own  advantages 
and  disadvantages.  For  most  purposes,  particularly  where  con- 
siderable power  is  to  be  transmitted,  the  Hindley  type  has  the 
advantage,  but  with  ordinary  machinery  it  is  somewhat  more 
difficult  to  obtain  the  same  degree  of  accuracy  as  can  be  obtained 
in  the  case  of  the  straight  type. 

From  tests  made  there  is  no  question  but  that  there  is  a  larger 
bearing  surface  on  the  Hindley  type  of  worm  than  on  the  straight. 

Therefore  this  type  of  gear- 
ing will  for  the  same  pitch 
present  a  bearing  of  greater 
durability,  and  heat  less  than 
the  straight  type,  particular- 
ly under  heavy  load.  The 
straight  type  may  have  less 
trouble  with  end-thrust  bear- 
ings. The  worm  can  move 
in  its  position  longitudinally 
with  the  worm  axis  and 
therefore  does  not  require  as 
close  an  adjustment  of  the 
end- thrust  bearings.  With 
first-class  bearings  the  Hind- 
ley  type  has  the  advantage,  as 

a  smaller  and  lighter  gear  can  be  used,  thus  reducing  the  expense. 
Some  efficiency  tests  on  an  ordinary  type  worm  and  worm- 
gear  for  automobile  rear-axle  drive,  for  electrical  vehicles  and 
light-power  cars,  were  undertaken  by  Mr.  Burgess.  A  trans- 
mission dynamometer,  similar  in  some  respects  to  the  apparatus 
used  at  the  Massachusetts  Institute  of  Technology  by  Professor 
Riley,  was  constructed.  The  prony  brake  was  adopted  for  an 
absorption  dynamometer,  and  a  long  shaft  of  small  diameter 
was  arranged  to  obtain  the  torsion  of  the  shaft  in  degrees  by  an 
electrical  indicator  apparatus  for  a  transmission  dynamometer. 
The  results  of  the  tests  are  given  in  the  accompanying  table. 


100 
95 
H   90 

UJ 

2   86 

LJ 

E 
>   80 

in    Hf 
o    75 

t 
u   70 

65 
60 

***^ 

^, 

f 

-«. 

>. 

/ 

/ 

i 

1 

J 

1 

1 

1 

1 

1234507 

BRAKE  HORSEPOWER    Machinery 

Fig.  3.  Diagram  showing  Relation  be- 
tween Brake  Horsepower  and  Efficiency, 
based  on  Results  of  Tests 


LOAD  AND  EFFICIENCY  183 

The  diagram,  Fig.  3,  gives  a  curve  plotted  from  the  results  ob- 
tained in  the  tests  and  recorded  in  the  table. 

Worm  Gearing  Employed  for  Freight  Elevators.  —  In  general 
the  worm  should  be  made  just  as  small  as  the  circumstances  will 
allow  in  order  to  increase  the  angle  of  thread  and  thereby  the 
efficiency,  while  maintaining  the  same  pitch  and  the  same  num- 
ber of  threads  on  the  worm. 

There  are  three  factors  which  may  determine  the  minimum 
size  of  worm  that  can  be  used,  which  are  as  follows:  First,  the 
diameter  of  the  shaft  on  which  the  worm  has  to  be  keyed,  if  not 
made  in  one  piece  with  this  shaft,  limits  the  size  of  the  worm. 
Second,  if  the  gear  is  to  be  self-locking  the  angle  of  the  thread 
cannot  be  increased  above  a  certain  degree;  with  the  pitch 
settled  on,  this  will  determine  the  diameter  of  the  worm,  provided 
it  is  single  threaded.  Third,  if  the  face  of  the  gear  is  determined, 
it  is  not  desirable  to  go  below  a  certain  diameter  of  worm  on 
account  of  the  consequent  large  face  angle. 

Factors  Determining  the  Load.  —  Concerning  the  load  which 
can  safely  be  carried  on  worm  gearing,  it  is  determined  by  one 
of  three  considerations,  which  are:  the  strength  of  the  material, 
the  danger  of  abrasion  and  the  danger  of  overheating. 

The  first  consideration  seldom  comes  into  play  because  a  gear 
proportioned  to  prevent  abrasion  and  excessive  heating  will 
generally  have  excessive  strength.  For  very  slow-running 
worms  and  for  worms  used  intermittently  with  short  runs  and 
long  intervals,  the  heating  effect  does  not  enter  and  the  deter- 
mining factor  will  be  the  danger  of  abrasion  from  too  high  a 
pressure  per  unit  of  contact  surface.  The  contact  between 
worm  and  worm-gear  is  mathematically  a  line,  but  the  physical 
properties  of  the  opposed  surfaces  and  the  lubricant  between 
them  expand  this  ideal  line  into  an  actual  area,  and  as  the  radii 
of  curvature  increase  directly  with  the  pitch,  it  is  natural  to  con- 
sider this  surface  as  directly  proportional  to  the  product  of  pitch 
and  face.  The  proper  allowable  load  per  unit  must  necessarily 
be  determined  by  experience,  and  1000  pounds  to  1200  pounds 
seems  to  be  about  the  safe  limit  of  load  per  unit  of  p  X  /  (pitch 
X  face)  considering  that  there  ought  to  be  here,  as  well  as  in  all 


184  WORM   GEARING 

other  designs,  a  certain  margin  or  factor  of  safety,  as  we  might 
say,  to  prevent  having  the  machine  put  out  of  commission  by 
an  occasional  overload  or  other  accidental  excessive  pressure.  If 
all  the  load,  as  is  usual  for  spur  gears,  is  considered  to  be  taken 
by  one  tooth,  the  stresses  produced  in  the  material  for  these 
loads  are  about  the  safe  stresses  for  cast  iron. 

Overheating.  —  The  third  consideration,  the  danger  of  over- 
heating, is  perhaps  the  most  important,  and  the  most  trouble 
with  worm  gearing  is  from  this  source.  When  more  heat  is 
developed  than  is  carried  off  from  the  gear  housing,  the  temper- 
ature of  the  oil  will  increase,  but  with  the  higher  temperature 
the  oil  becomes  less  viscous  and  its  adhesion  to  the  rubbing  sur- 
faces becomes  less.  The  coefficient  of  friction  increases  with  con- 
sequent more  rapid  increase  in  temperature.  Thus  the  critical 
conditions  are  constantly  augmented  by  one  another  until  the 
oil  film  between  the  surfaces  is  squeezed  out  altogether  and 
abrasion  occurs.  The  only  safe  way  to  avoid  this  is,  of  course, 
to  so  design  the  gearing  that  the  temperature  is  kept  below  a 
certain  limit.  For  continuous  service  the  proper  loads  may 
be  based  on  Bach's  and  Roser's  experiments  referred  to  in  the 
preceding  sections  of  this  chapter.  It  may  be  well  here  to  call 
attention  to  the  fact  that  the  loss  of  heat  from  a  body  is  approxi- 
mately directly  proportional  to  its  surface,  and  consequently  a 
large  gear  housing  is  at  an  advantage.  The  housing  should 
have  stuffing-boxes  for  the  worm  shaft,  and  be  well  filled  with  a 
viscous  oil  so  that  the  heat  created  at  the  point  of  contact  may 
be  distributed  quickly  to  the  upper  parts  of  the  housing. 

Special  Application  to  Freight  Elevators.  —  For  intermittent 
duty,  like  that  imposed  on  a  freight  elevator,  the  question  of 
allowable  load  becomes  more  complicated.  The  load  that  can 
safely  be  carried  on  a  gear  for  this  class  of  work  will  depend  en- 
tirely on  the  circumstances,  and  a  value  can  only  be  arrived  at 
if  these  are  known,  or  after  certain  assumptions  as  to  the  maxi- 
mum time  of  continuous  service,  time  of  intervals,  etc.,  have  been 
made.  The  total  heat  developed  can  then  be  compared  to  that 
for  continuous  service  and  a  correspondingly  higher  load  allowed. 

Consider,  for  instance,  a  worm  for  driving  a  freight  elevator 


LOAD   AND   EFFICIENCY  185 

with  a  load  on  a  24-inch  drum  of  4000  pounds,  worm  direct  on 
motor  shaft  running  at  850  R.P.M.  If,  in  this  instance,  it  is  con- 
sidered safe  to  assume  that  the  maximum  average  load  for  a 
certain  unit  of  time  will  never  exceed  2000  pounds  and  the  time 
required  for  loading  and  unloading  the  elevator  is  at  least  equal 
to  the  time  of  actual  running,  then  the  work  performed  by  the 
worm  gearing  will  be  one-fourth  of  that  for  continuous  service 
with  full  load,  assuming  the  coefficient  of  friction  the  same  for 
all  loads.  The  heat  developed  will  also  be  one-fourth  of  that 
developed  with  full  load.  A  gearing  designed  for  continuous 
service  with  1000  pounds  load  on  drum  will  therefore  meet  the 
requirements. 

For  a  worm  of  3!  inches  in  diameter  running  at  850  R.P.M.  , 
we  have  a  velocity  of  824  feet  per  minute.  For  this  velocity  a 
load  per  unit  of  pitch  times  face  of  180  pounds  is  allowable  for 
a  difference  in  temperature  of  50  degrees  F.  The  gear  will  have 
to  have  108  teeth  to  give  the  necessary  reduction  to  50  feet 
per  minute  elevating.  As  this  number  of  teeth  is  exceptionally 
large,  we  can  expect  a  good  contact  with  less  danger  of  abrasion, 
and  a  higher  temperature  difference,  say,  70  degrees,  is  warranted. 
The  allowable  load  is  approximately  proportional  to  the  temper- 
ature difference,  and  we  can,  therefore,  allow  180  X  f-§-  =  252 
pounds  per  unit  of  pitch  times  face.  On  account  of  the  large 
diameter  of  the  worm-gear,  the  worm  and  its  housings  will  be 
comparatively  long  with  consequent  large  radiating  surface,  and 
a  temperature  difference  of  70  degrees  will  probably  not  be 
reached  at  all. 

A  worm,  4  inches  in  diameter,  i  inch  pitch,  with  gear  34.4 
inches  in  diameter,  i\  inches  face,  will  then  carry  252  X  i  X  2f  = 
693  pounds,  which  corresponds  to  a  load  on  the  drum  =  693  X 


=  993   pounds,   or  practically   1000  pounds.     The  inter- 

24 

mittent  load  on  gears  will  be  4000  X  —  —  =  2791  pounds,  or 

34-4 

1015  pounds  per  (p  X  /),  which  in  this  case  is  within  the  limit. 

Frequently  Employed  Objectionable  Designs.  —  Many  worm- 
gearing  designs  used  on  freight  elevators  employ  too  high  a  load 


186  WORM  GEARING 

on  the  gear  teeth.  In  one  case  a  worm-gear  having  108  teeth, 
|  inch  pitch,  2^  inches  face  and  a  worm  5T3g  inches  in  diameter, 
single  threaded,  was  used.  The  worm  was  direct-connected  to 
an  electric  motor  running  at  850  revolutions  per  minute.  Diffi- 
culties were  experienced  with  regard  to  the  heating  of  the  worm. 
The  current  required  by  the  motor  was  also  too  great.  The 
winding  drum  was  24  inches  in  diameter  and  the  load  on  the  drum 
4000  pounds,  which  corresponds  to  3720  pounds  on  the  worm- 
gear  teeth. 

With  the  dimensions  given,  the  angle  of  thread  is  2  degrees 
39  minutes,  and  the  efficiency  of  the  worm-gearing  for  a  coeffi- 
cient of  friction  equal  to  0.05  would  be  0.48  (see  preceding  sec- 
tions of  this  chapter).  If  the  diameter  of  the  worm  were  reduced 
to  3}  inches,  the  angle  of  the  thread  would  be  increased  to  3  de- 
grees 39  minutes  and  the  efficiency  to  0.58,  an  increase  in  efficiency 
of  21  per  cent.  It  will  be  seen  from  this  that  a  decrease  in  worm 
diameter  not  only  reduces  the  speed  of  the  rubbing  surfaces,  but 
also  increases  the  efficiency. 

A  load  on  the  gear  teeth  of  3720  pounds  for  a  f-inch  pitch, 
2^-inch  face  gear,  corresponds  to  a  load  per  unit  of  p  X  /  equal 
to  1984  pounds.  This  is  without  question  too  heavy  a  load, 
even  for  intermittent  service,  and  worm  gearing  with  any  such 
load,  running  at  high  speed,  is  likely  to  give  trouble.  Upon  being- 
advised  that  the  gears,  as  described,  were  too  small  for  the  ser- 
vice required  of  them,  the  manufacturers  of  the  gearing  stated 
that  they  were  building  elevators  in  competition  with  other  con- 
cerns and  that  a  material  increase  in  these  gears  would  make  it 
impossible  for  them  to  compete  successfully.  They  also  stated 
that  they  had  sometimes  operated  a  load  of  6000  pounds  with  a 
lohorsepower  motor,  but  in  one  or  two  cases  they  had  found  it 
very  difficult  to  start  the  elevator  except  by  using  a  heavy  current. 

Now  6000  pounds  at  50  feet  per  minute  represents  -^^  -  — 


=  9.1  H.P.  The  efficiency  of  a  worm-gear  with  an  angle  of 
thread  2  degrees  40  minutes  was  found  above  to  be  0.48  for  a 
coefficient  of  friction  of  0.05.  This  is  the  efficiency  of  the  worm 
gearing  itself  and  does  not  allow  for  friction  in  gear  or  worm- 


LOAD  AND  EFFICIENCY 


shaft  bearings,  for  end-thrust  bearing,  for  bending  of  cables  or 
friction  in  guides.  When  all  this  is  taken  into  consideration  the 
horsepower  required  for  running  conditions  will  be  at  least  22. 
The  horsepower  for  starting  will  be  still  higher  and  a  correspond- 

Horsepower  Transmitted  by  Worm  Gearing,  Single-threaded  Worm 

(Gear:  Phosphor-bronze;  Worm:  Hardened  Steel) 


Single-threaded 

Single-threaded 

-_ 

Single-threaded 

Worm 

Worm 

Worm 

Horse- 

Horse- 

Horse- 

power 

> 

1 

power 

8 

power 

— 

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to  be 
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ing  electric  current  consumption  in  the  motor  must  necessarily 
result,  which  indeed  must  be  called  high  for  a  zo-H.P.  motor. 

Lubricant  for  Worm-gears.  —  In  the  majority  of  installations 
the  worms  are  cut  from  steel  while  the  worm-wheels  are  of  cast 
iron.  A  very  satisfactory  lubricant  is  composed  of  the  follow- 
ing ingredients:  Cylinder  oil,  2  gallons;  common  flour,  i  pound; 
common  salt,  \  pound. 

It  will  be  found  that  the  flour  will  make  the  oil  heavy  enough 


i88 


WORM   GEARING 


to  stick  while  the  salt  will  so  glaze  the  worm  and  gear  that  they 
will  run  smoothly  without  any  tendency  to  "score."  In  ex- 
treme cases,  where  the  worm-gear  runs  hot,  owing  to  continuous 
and  fast  running  and  to  friction,  it  is  sometimes  advisable 
to  add  to  the  above  ^  pound  of  graphite.  The  lubricating  and 

Horsepower  Transmitted  by  Worm  Gearing,  Double-threaded  Worm 

(Gear:   Phosphor-bronze;  Worm:  Hardened  Steel) 


Double-threaded 

Double-threaded 

Double-threaded 

Worm 

Worm 

Worm 

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Horse- 

Horse- 

power 

tr* 

g 

power 

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cooling  properties  of  graphite  are  too  well-known  to  require  dis- 
cussion. Anyone  who  has  had  trouble  with  worm  gearing  will 
find  this  lubricant  well  worth  trying. 

Horsepower  of  Worm  Gearing.  —  A  great  many  manufacturers 
of  worm-gear  drives  in  Europe  build  and  guarantee  drives  for 
a  given  horsepower  at  a  given  speed.  The  accompanying  tables 
give  the  horsepower  that  may  be  safely  transmitted  by  worm 
gearing  at  given  speeds  of  the  worm.  It  is  important  in  worm 


LOAD  AND  EFFICIENCY 


189 


drives  to  keep  the  diameter  of  worm  .as  small  as  possible.  The 
tables,  therefore,  give  the  diameter  of  the  largest  allowable  pitch 
diameter  of  the  worm.  It  is  not  advisable  to  make  the  worm 
larger,  as  the  gearing  may  then  run  hot  and  start  to  cut,  but  there 
is  no  objection  to  making  the  pitch  diameter  smaller,  if  the 

>^_____, —i     ,          -      '          .1        ^"""^  *  • 

Horsepower  Transmitted  by  Worm  Gearing,  Triple-threaded  Worm 

(Gear:   Phosphor-bronze;   Worm:  Hardened  Steel) 


Triple-threaded 

Triple-threaded 

Triple-threaded 

Worm 

Worm 

Worm 

Horse- 

Horse- 

Horse- 

power 

,     8 

power 

8 

power 

r! 

I 

to  be 
Trans- 
mitted 

Rev.  per 

Minute 
of  Worm 

Linear  Pitc: 
of  Worm, 
Inches 

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mitted 

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linear  pitch  of  the  worm  is  not  too  coarse  to  prevent  this.  In 
many  cases  in  the  tables  the  pitch  is  so  coarse  with  relation  to 
the  pitch  diameter  that  it  would  not  be  possible  to  have  a  separ- 
ate worm  mounted  on  a  shaft,  but  the  worm  must  be  made  an 
integral  part  of  the  worm-shaft.  Where  there  is  a  choice  between 
using  single,  double  or  triple  threads  for  the  same  drive,  it  is 
preferable  to  use  double  or  triple  threads.  The  tables  apply  to 
phosphor-bronze  worm-wheels  and  hardened  steel  worms.  If 
the  worm-wheel  is  made  of  cast  iron,  instead  of  phosphor-bronze, 
the  pitch  should  be  made  if  greater  than  that  given  in  the  tables. 


igo 


WORM  GEARING 


In  the  case  of  unfinished  teeth,  the  pitch  diameter  of  the  worm 
should  be  only  0.8  times  that  given  in  the  tables. 

The  best  material  for  worm  gearing  is  hard  phosphor-bronze 
for  the  worm-wheel  and  hardened  steel  for  the  worm.  The 
next  best  materials  are  cast  iron  for  the  worm-wheel  and  hard- 
ened steel  pr  cast  iron  for  the  worm.  Steel  or  steel  castings  for 

Horsepower  Transmitted  by  Worm  Gearing,  Quadruple-threaded  Worm 

(Gear:  Phosphor-bronze;  Worm:  Hardened  Steel) 


Quadruple- 

Quadruple- 

Quadruple- 

threaded  Worm 

threaded  Worm 

threaded  Worm 

Horse- 

Horse- 

Horse- 

power 
to  be 
Trans-* 

|f| 

Is* 

111 

power 
to  be 
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both  the  worm-wheel  and  worm  are  only  allowable  for  slow 
speeds.  The  teeth  in  the  worm-wheel  and  the  thread  on  the 
worm  should  always  be  cut,  whenever  the  gearing  is  to  be  used 
steadily  or  at  a  reasonably  high  speed. 

These  tables  are  based  upon  the  practice  of  a  prominent 
European  manufacturer  making  worm  drives  guaranteed  to 
transmit  given  amounts  of  power.  Information  on  this  subject 
is  extremely  scarce  and  no  tables  have  previously  been  published 
in  this  form  giving  the  horsepower  transmitted  by  worm  gearing, 
except  in  MACHINERY'S  HANDBOOK,  from  which  these  tables  are 
reproduced. 


CHAPTER  VIII 
THE  DESIGN   OF   SELF-LOCKING  WORM-GEARS 

THE  old  opinion  that  the  friction  and  wear  of  worm-gears  are 
necessarily  very  great,  and  that  the  efficiency  is  necessarily  very 
low,  making  worm  gearing  an  unmechanical  contrivance,  is  not 
as  frequently  met  with  now  as  formerly.  Unwin  states  that  in 
well-fitted  worm  gearing,  of  speed  ratios  not  exceeding  60  or  80 
to  i,  motion  will  be  transmitted  backwards  from  the  wheel  to  the 
worm.  In  Prof.  Forrest  R.  Jones'  work  on  machine  design  may 
be  found  tabulated  the  results  of  many  examples  from  practice, 
some  of  which  show  an  efficiency  as  high  as  74  per  cent  before 
abrasion  began,  the  most  notable  example  being  that  of  a  worm 
running  at  a  surface  speed  of  306  feet  per  minute  under  a  load  of 
5558  pounds,  and  showing  an  efficiency  of  67  per  cent,  with  no 
abrasion.  The  tables  in  Professor  Jones'  work  show  that  under 
light  loads  very  high  surface  or  rubbing  speeds  are  allowable, 
running  as  high  as  800  feet  per  minute.  It  has  also  been  pointed 
out  that  an  increase  in  the  thread  angle,  in  general,  increases  the 
efficiency. 

There  is,  however,  an  important  function  of  worm  gearing 
which  is  not,  as  a  rule,  brought  out  adequately  by  writers  on 
worm  gearing,  and  which  in  certain  classes  of  machinery  is  of 
the  first  importance,  often,  indeed,  becoming  the  deter- 
mining factor  in  deciding  upon  the  choice  of  a  worm-gear 
as  the  power  transmitter.  It  is  the  property  a  worm-gear 
possesses,  under  certain  conditions  dependent  upon  its  de- 
sign, of  being  self-locking,  and  preventing  motion  back- 
wards. 

An  instance  where  this  property  becomes  of  prime  importance 
and  accounts  for  the  use  of  the  worm-gear  is  in  crane  work,  where 
the  winding  drum  is  driven  by  a  worm-gear  so  designed  that, 
when  the  power  is  shut  off,  the  gear  will  not  run  down  or  back- 

191 


WORM   GEARING 


wards  under  the  impulse  of  the  load,  but  will  be  self-locking, 
holding  the  load  at  any  point. 

Fig.  i  shows  a  single-thread  worm  in  mesh  with  the  worm- 
wheel,  a  being  the  angle  of  the  worm  thread  with  the  axis  of  the 
worm-wheel,  and  in  order  that  the  system  may  be  self-locking, 
that  is,  that  the  worm-wheel  may  be  unable  to  run  the  worm, 
the  tangent  of  the  angle  a  must  be  less  than  the  coefficient  of 
friction  between  the  teeth  of  the  worm  and  wheel,  or  as 

tana  =  —.,    so       j<f  (i) 

ird  ird 

in  which  p  =  the  pitch;  d  =  the  pitch  diameter  of  the  worm; 
and  /  =  the  coefficient  of  friction  between  the  worm  and  wheel. 


Machinery 


Fig.  i.     Single-threaded  Worm  in  Mesh  with  Worm-wheel,  used  to 
Illustrate  the  Principle  of  Self-locking  Worm  Gearing 

It  is  necessary  to  assume  a  value  for/,  which,  if  the  condition  of 
determining  the  use  of  the  worm-gear  is  its  self-locking  property, 
should  be  assumed  conservatively  low.  Unwin  states  under  the 
authority  of  Professor  Briggs  that  a  well-fitted  worm-gear  will 
exhibit  a  lower  coefficient  of  friction  than  any  other  kind  of 
running  machinery.  Professor  Jones  gives  a  series  of  values  for 
the  coefficient  of  friction  of  screw  gears,  one  of  which  is  a  pinion 
of  4  inches  pitch  diameter,  the  average  value  being  /  =  0.05, 
corresponding  to  a  rubbing  velocity  of  250  feet  per  minute. 
Mr.  Halsey  assumes/  =  0.05,  and  Mr.  Wilfred  Lewis  says  that 
when  the  worm-gear  is  worked  up  to  the  limit  of  its  safe  strength, 


SELF-LOCKING  WORM-GEARS  193 

a  rubbing  velocity  greater  than  200  to  300  feet  per  minute  will 
prove  bad  practice.  It  is  in  heavy  machinery  where  worm- 
gears  are  mostly  used  as  self-locking  transmission  elements,  and 
here  they  are  usually  worked  up  to  the  safe  strength  of  the 
wheel;  hence  it  is  fair  to  assume/  =  0.05  when  designing  a  self- 
locking  worm-gear,  and  to  limit  the  rubbing  velocity  to  200  feet 
per  minute,  and  we  have  for  the  limiting  value  of  p  at  which 
the  system  will  be  self-locking: 

p  =  0.05  ird  =  0.157  d  (2) 

The  sliding  velocity  in  feet  per  minute  at  the  pitch  line  is  ex- 
pressed by 

T.     TT  dn  ,  N 

V= =  0.262  dn  (3) 

where  d  =  the  pitch  diameter  of  the  worm,  and  n  =  the  number 
of  revolutions  per  minute  of  the  worm. 

Under  the  above  assumption,  that  for  continuous  service  and 
heavy  pressures  the  sliding  velocity  should  not  be  more  than  200 
feet  per  minute,  we  have  as  the  limiting  value  of  d  to  avoid  all 
cutting: 

d_     200 
0.262  n 

The  exact  nature  of  the  surface  of  contact  between  a  worm 
and  wheel  is  involved  in  doubt;  many  claim  it  is  only  a  point; 
it  certainly  is  not  large,  and  consequently  a  wide  face  for  the 
wheel  is  not  needed. 

If  the  angle  <zi  is  made  60  or  75  degrees,  it  will  make  the  face 
satisfactory  for  any  ordinary  worm  of  from  4  to  6  inches  in  diam- 
eter. 

There  is  in  all  worm  gearing  a  very  heavy  end-thrust  on  the 
worm-shaft,  and  also  an  outward  force  normal  to  the  worm-axis, 
each  of  which  must  be  suitably  provided  for  in  the  design  of  the 
shaft  and  bearings.  The  end-thrust  may  be  taken  by  bronze 
washers  slipped  into  the  bearings  at  the  end  of  the  shaft,  which 
may  be  removed  when  worn  and  replaced  with  new  ones.  Shoul- 
ders may  be  provided  on  the  shaft,  between  which  and  the  bear- 
ings bronze  collars  may  be  placed,  these  being  split  to  enable  new 


IQ4 


WORM   GEARING 


ones  to  be  easily  and  quickly  placed  in  position  when  the  old 
ones  become  worn.  Roller  thrust  bearings  are  very  often  applied 
to  worms,  and  these  as  well  as  the  bronze  washers  may  be  sup- 
plied with  adjusting  set-screws  to  take  up  the  wear,  instead  of 
renewing  the  washers. 

In  Fig.  2  let  P  =  the  tangential  force  at  the  pitch  line  of  the 
worm,  d  =  the  pitch  diameter  of  the  worm,  Q  =  the  tangential 
force  at  the  pitch  line  of  the  worm-wheel,  E  =  the  end-thrust  of 


Machinery 


Figs.  2  and  3.    Diagrams  for  the  Derivation  of  Formulas 

the  worm-shaft,  and  F  =  the  force  on  the  worm-shaft  normal 
to  the  worm-axis;  then,  friction  being  neglected: 


(4) 


In  Fig.  3  draw  line  EC  parallel  to  the  axis,  or  coinciding  with 
the  pitch  line,  of  the  worm ;  let  this  line  represent  the  force  E  = 
Q\  draw  AB  normal  to  this  line;  it  will  then  also  be  normal  to 
the  axis  of  the  worm ;  then,  when  measured  to  the  same  scale  to 
which  BC  is  drawn,  AB  =  F;  if  the  angle  CAB  is  75!  degrees, 
we  have: 


-  =  tan  14^  degrees 
F  =  0.250  Q 


(5) 
(6) 


Taking  friction  into  consideration,  the  force  PI,  tangential  to 
the  pitch  line  of  the  worm,  which  it  is  necessary  to  employ  in 


SELF-LOCKING  WORM-GEARS  195 

order  to  produce  a  force  Q  tangential  to  the  pitch  line  of  the 
wheel,  is  given  by  Weisbach  as 


(7) 

A    —    ftj 

in  which 


The  efficiency  of  the  worm  and  wheel  is  then 


Example.  —  A  single-thread  worm  of  i-inch  pitch,  running 
80  revolutions  per  minute,  is  to  transmit  to  a  worm-wheel  a 
tangential  force  Q  =  5000  pounds,  and  is  to  be  self-locking. 

From  (3) 

200 


d< 


0.262  X  80 


or  d  may  be  as  large  as  9.5  inches  before  abrasion  need  be  feared. 

From  (2),  p  <  0.157  J;  assume  p  =  0.125  d;  then,  as  p=  i 
inch,  d  =  8  inches,  or  the  worm  will  require  to  be  8  inches  pitch 
diameter  in  order  that  the  angularity  of  the  thread  may  be  small 
enough  to  make  the  system  self-locking.  It  will  be  seen  that  the 
required  diameter  will  be  increased  as  the  value  of  /  is  decreased, 
and  in  case  the  required  diameter  of  the  worm  proves  too  great 
for  practice,  and  the  pitch  cannot  be  reduced  on  account  of  con- 
siderations of  strength,  some  outside  aid,  such  as  a  brake  or 
friction  disk  applied  to  the  worm-shaft,  will  have  to  be 
adopted. 

From  (7)  as 

7       P  i 

h  =  J~1  =  —   —  -  =  0.04 

ird      3.14  X  8 
we  have 


From  (4) 


n        0.04  +  0.05  , 

PI  =  5000  -  -^  —  vx  J  .  =45i  pounds. 
i  -  (0.04  X  0.05) 


5000  =  ^4  -  ,  or  P  =  199  pounds. 


WORM   GEARING 


From  (8) 


=  44  per  cent  for  the  efficiency  of  the  worm-gear. 


Pi      45i 

The  formulas  may,  by  starting  with  those  for  the  efficiency,  be 
used  to  determine  the  pitch  diameter  which  will  give  the  proper 
thread  angle  for  any  given  pitch  and  degree  of  efficiency. 

It  is  clear  from  the  foregoing  that  a  worm-gear  of  large  pitch 
will  require  a  pitch  diameter  of  the  worm  altogether  too  large  for 
practice,  if  it  is  to  be  self-locking,  and  that  the  system  as  usually 
designed  may  be  expected  to  run  backwards.  To  prevent  this 
a  friction  disk  may  be  placed  in  the  bearing  which  receives  the 
thrust  of  the  worm-shaft  when  the  system  is  running  backwards, 
and  the  diameter  of  the  disk  so  proportioned  as  to  just  hold 
the  worm-shaft  stationary  under  the  impulse  of  the  worm- 
wheel. 


Machinery 


Figs.  4  and  5.     Diagrams  for  Determining  Bearing  Friction 

Bearing  Friction.  —  The  foregoing  discussion  neglects  the 
effect  of  the  thrust  of  the  worm-shaft  in  its  bearings,  the  frictional 
resistance  of  which  must  be  added  to  that  of  the  teeth  to  obtain 
the  actual  conditions  of  a  self-locking  system.  This  frictional 
resistance  depends  upon  the  values  of  the  end-thrust  E  and 
the  normal  force  F  already  found,  and  the  diameter  and  form  of 
the  bearing.  In  nearly  all  cases  of  worm  gearing  the  mounting 
of  the  worm  upon  the  shaft  will  be  covered  by  one  of  three  cases, 
either  unsymmetrically  between  the  bearings,  symmetrically 
between  the  bearings,  or  overhung. 

In  Case  i,  Fig.  4,  the  bending  moment  upon  the  worm-shaft  is 


(9) 


TUT   —  — 


SELF-LOCKING  WORM-GEARS 


197 


In  Case  2,  same  as  Case  i,  except  that  the  worm  is  central 
between  the  bearings,  and 


the  bending  moment  upon  the  worm-shaft  is 

M  =  0.250  QL 

4 

In  Case  3,  Fig.  5,  the  bending  moment  upon  the  worm-shaft  is 

M  =  FL  =  0.250  QL.  (n) 

In  each  of  the  above  cases  the  shaft  is  subjected  to  a  combined 

twisting  and  bending  strain,  the  twisting  moment  being  the  same 

in  each  case,  T  =  PR,  which  is,  however,  so  small  as  to  be 

negligible  in  what  follows. 

In  the  following  table  the  first  column  shows  the  several  styles 
of  journals  most  commonly  used  for  worm-shafts,  the  second 

Table  giving  Moment  of  Friction  with  Various  Types  of  Bearings 


Style  of  Journal 


Moment  of  Friction 


Moment  of  Friction 


fFd* 

2 


0.05 


0.04  Pdz 
P 


2fEr 
3 


0.05 


0.2  Pr  _  o.i  Pdz 
P  P 


C       Ys 


3  r  sin  y 


0.05 


0.2  P(r3  -rtf 
pr  sin  y 


0.05 


o.2P(n3-r3) 
ri2  -  r2 


column  gives  the  moment  of  friction  for  each  under  a  load  in  the 
direction  of  the  arrow,  the  third  column  gives  the  coefficient 
of  friction  assumed,  and  the  fourth  column  gives  the  tangential 
force  P2  at  the  pitch  line  of  the  worm,  resulting  from  the  resist- 


i98 


WORM   GEARING 


ance  of  friction  in  the  journals,  and  found  by  dividing  the  mo- 
ment of  friction  in  Column  2  by  the  pitch  radius  of  the  worm. 

There  are  always  acting  upon  the  worm-shaft  the  two  forces  F 
and  E\  consequently  to  get  the  resultant  retarding  force  tan- 
gential to  the  pitch  line  of  the  worm,  we  must  take  the  sum  of 
the  resultants  due  to  the  frictional  resistance  of  each  force  sepa- 
rately. Referring  to  the  table,  for  each  worm-shaft,  find  the  con- 
ditions shown  at  A,  in  addition  to  the  conditions  shown  either  at 
B,  C  or  D,  as  the  case  may  be,  and  the  total  resultant  force  PZ 
at  the  worm  pitch  line  will  be  the  sum  of  the  quantities  given  in 
Column  4  opposite  the  particular  cases. 


Machinery 


Fig.  6. 


Diagram  for  Determining  the  Angle  of  Repose  Correspond- 
ing to  the  Journal  Friction 


These  frictional  resistances  developed  by  the  journals  act  in  a 
direction  helpful  to  the  self-locking  property  of  the  worm,  and 
enable  the  designer  to  use  a  larger  thread  angle  for  a  given  diam- 
eter of  worm,  or  a  smaller  diameter  of  worm  for  a  given  thread 
angle,  thus  keeping  within  the  limits  of  good  practice,  and  in- 
creasing the  efficiency  of  the  system  for  the  forward  movement. 

Having  determined  the  force  P2  tangential  to  the  worm  pitch 
line,  resulting  from  the  frictional  moment  at  the  journals,  the 
angle  of  repose  for  this  force  acting  with  the  force  Q,  as  shown  in 
Fig.  6,  is  given  by  the  equation, 

P2 

tan  x  =  — 

The  thread  angle  found  previous  to  the  consideration  of  the 
effect  of  the  journal  friction  may  now  be  increased  by  the  angle 


SELF-LOCKING  WORM-GEARS 


IQ9 


x}  making  the  thread  angle  a  +  x.  This  may  be  accomplished 
either  by  increasing  the  thread  angle,  increasing  the  pitch,  or 
decreasing  the  pitch  diameter. 

Consider,  now,  that  in  the  foregoing  example,  the  worm-shaft 
is  of  the  form  in  Case  2,  the  worm  being  central  between  the 
bearings,  and  the  distance  between  bearings  being  36  inches. 

Then,  from  (5),  we  have: 

F  =  0.250  X  5000  =  1250  pounds. 
From  (10) 

,r      0.250  X  5000  X  36  .     , 

M  =  -  — *-  =  11,250  inch-pounds. 

4 

Assuming  s  =  10,000  pounds  per  square  inch  for  the  allowable 
fiber  stress  in  the  worm-shaft,  we  have: 


M  =  ^ 
32 


or 


2.28  inches. 


From  the  table,  Case  A, 

n       0.04  X  199  X  2.28 

PI  =  -  — za =  18.15  pounds. 

From  the  table,  Case  B, 

„       o.i  X  199  X  2.28  , 

Pi  =  -          « =  45-37  pounds. 


Then 


P2  =  18.15  +  45.37  =  63.52  pounds. 
2 


From  (i) 


tan#  =  =  0.0127 

5000 

x  =  o  deg.  44  min. 


tana 


0.04 


V.i4  X  8 
a  =  2  deg.  17  min. 


Then 


a  +  x  =  3  deg.  i  min. 
tan  3  deg.  i  min  =  0.053 

f-  =  0.053  =  h,  and  d  =  6  inches,  approx. 
ird 


200  WORM   GEARING 

If,  now,  we  substitute  these  new  values  of  h  and  d  in  Equations 
(7)  and  (4),  we  continue  as  follows: 
From  (7) 


From  (4) 


„  0-053  +  °-°5  /:  j 

PI  =  5000 T-**-        — ~ — r  =  5l6  pounds. 

i  -  (0.053  X  0.05) 


'P  X  3.14  X6  , 

5000  =  -  — -  ,   or   P  =  265  pounds. 

From  (8) 

—  =  — ^  =  51  per  cent  efficiency  for  the  worm-gear. 
PI      5l6 

The  total  efficiency  of  the  system,  taking  account  of  the  journal 
friction,  will  be: 

P  265 

7T~r^7  =  — 7~~>          -  =  46  per  cent. 
Pi  +  P2      516  +  63.52 

It  thus  becomes  clear  that  while  the  efficiency  of  the  worm 
threads  and  wheel  teeth  has  been  increased  above  50  per  cent, 
the  efficiency  of  the  whole  system,  including  the  journals,  is 
below  50  per  cent,  and  the  system  retains  its  self-locking  prop- 
erty. It  is  evident  that  when  running  forward,  the  end-thrust 
E  upon  the  worm-shaft  will  be  upon  the  opposite  end  from  that 
when  running  backward,  and  on  this  account  a  system  may  be 
designed  to  have  a  high  efficiency  on  the  forward  movement  and 
still  preserve  its  self-locking  property. 

If  both  the  journals  have  roller  bearings,  and  the  end  taking 
the  thrust  on  the  forward  movement  has  a  ball  bearing,  while 
the  opposite  end  be  made  like  Case  C  or  D  in  the  table,  properly 
proportioned,  the  worm  may  be  designed  to  show  a  very  high 
efficiency  on  the  forward  movement,  while  the  frictional  resist- 
ance of  the  step  bearing  on  the  opposite  end  will  cause  the  sys- 
tem to  be  self-locking  by  reason  of  the  energy  absorbed  at  the 
step  bearing. 

General  Method  of  Procedure.  —  The  formulas  may  be  put 
into  more  convenient  form  for  this  purpose,  as  follows:  The  de- 
signer will  have,  to  start  with,  a  knowledge  of  the  force  Q  re- 
quired at  the  worm-wheel,  the  force  PI  at  the  pitch  line  of  the 
worm,  developed  from  the  source  of  power,  the  pitch  required 


SELF-LOCKING  WORM-GEARS  2OI 

for  the  worm-wheel,  and  the  efficiency  e  for  which  he  wishes  to 
design  the  system.     We  then  have: 

p 

—  •  =  e,   and  P  =  P\e. 

"i 

Substituting  this  value  for  P  in  Equation  (4)  and  solving  for 
d,  we  have  : 


3. 

for  the  worm,  neglecting  the  journals,  when  the  journals  and 
thrust  bearings  are  roller  and  ball  bearings,  respectively,  and 

d_         PQ 

3.14  (Pi  -  P2)  e 

when  the  journals  and  thrust  bearings  are  considered. 

The  worm  being  thus  designed  for  the  given  efficiency  on  the 
forward  movement,  it  remains  to  determine  such  proportions  of 
the  step  bearing  for  the  backward  movement  as  will  present 
enough  frictional  resistance  to  render  the  system  self-locking. 
Let  ei  =  the  efficiency  when  the  journals  and  thrust  are  con- 
sidered, then: 


or        = 


r\  +  "2 

and  substituting  the  value  of  P  found  above, 
ePl  = 


and 

Pi  (e  - 


By  equating  this  force  P2  to  the  proper  quantity  from  Column 
4  in  the  table  of  journal  resistances,  the  proportions  required  of 
the  journal  or  step  bearing  may  be  determined. 


CHAPTER  IX 
THE  HINDLEY  WORM  AND   GEAR 

THE  Hindley  type  of  worm-gear  was  first  used  in  Hindley's 
dividing  engine,  and  was,  by  the  inventor,  considered  superior 
to  the  ordinary  type,  in  wearing  quality.  Investigation  has 
practically  settled  that  the  nature  of  contact  between  the  worm 
thread  and  the  teeth  of  the  ordinary  worm-wheel  is  that  of 
line  contact,  extending  across  the  tooth  on  the  pitch  line.  It 
has  also  been  fairly  well  proved  in  practical  examples  that  the 
contact  is  of  a  broader  nature  on  account  of  the  elasticity  of  the 
materials  used  in  the  construction.  The  convex  surfaces  of 
contact  are  flattened  considerably  under  pressure  and  thus  for 
practical  purposes  make  actual  surface  contact.  The  contact 
in  the  ordinary  worm  and  worm-wheel  type  is  limited  to  two 
teeth  of  the  wheel  and  worm  thread,  at  most. 

Comparison  of  Ordinary  and  Hindley  Worm  Gearing.  —  The 
conditions  are  much  different  in  the  case  of  the  Hindley  worm, 
and  it  is  the  intention  in  this  chapter  to  show  wherein  the  differ- 
ence lies.  As  this  style  of  gearing  is  not  very  common,  a  few 
words  regarding  its  construction  will  not  be  out  of  place.  Fig.  i 
illustrates  the  Hindley  worm,  showing  the  theoretical  form. 
This  worm  is  not  of  cylindrical  shape,  but  is  formed  somewhat 
like  an  hour-glass,  after  which  it  is  sometimes  named.  The 
worm  blank,  being  made  smaller  in  diameter  in  the  middle  than 
at  either  end,  conforms  to  the  circumference  of  the  wheel  with 
which  it  meshes.  The  worm  thread  is  cut  by  a  tool  which  moves 
in  a  circular  path  about  a  center  identical  with  the  axis  of  the 
wheel  with  which  it  is  to  mesh,  and  in  the  plane  in  which  the 
axis  of  the  worm  lies.  The  process  is  similar  to  ordinary  thread 
cutting  in  the  engine  lathe,  except  for  the  difference  in  the  path 
of  the  tool,  the  tool  having  a  circular  instead  of  a  straight 
path. 

202 


HINDLEY  WORM  AND   GEAR 


203 


It  is  evident  that  the  worm  shape  is  dependent  on  the  partic- 
ular wheel  with  which  it  is  to  run,  and  Hindley  worms  are  not 
interchangeable  with  any  other  but  an  exact  duplicate.  That  is, 
a  worm  cut  for  a  Hindley  gear  of  50  teeth  cannot  be  used  success- 
fully with  a  wheel  of  70  teeth,  although  the  pitch  of  the  teeth  is 
exactly  the  same.  In  the  ordinary  type  of  worm  gearing  one 
worm  may  be  made  to  run  with  any  number  of  diameters  of 
wheels  of  the  same  pitch,  and  bobbed  with  the  same  hob. 


Machinery 


Fig.  i.     Typical  Hindley  Worm 

In  action  the  two  styles  of  worm-gear  differ  greatly,  and  both 
diverge  widely  in  action  from  the  case  of  a  plain  nut  and  screw, 
which  may  be  taken  to  represent  a  worm  and  worm-gear,  the 
latter  of  infinite  diameter  and  with  an  angle  of  embrace  of  360 
degrees.  In  studying  the  action  between  the  thread  and  teeth 
of  the  ordinary  type  of  worm-gear,  we  must  understand  odontics, 
rolling  contacts  and  the  theory  of  tooth  gearing  in  general,  in 
order  to  understand  the  action  of  the  ordinary  worm-gear.  But, 
in  studying  the  action  of  the  Hindley  type,  we  are  concerned  with 
no  such  theories,  as  the  action  is  purely  sliding  and  devoid  of 
rolling  contact.  In  the  ordinary  worm  we  have  an  axial  pitch 


204 


WORM   GEARING 


which  is  constant  from  top  to  root  of  the  thread,  while  in  the 
Hindley  worm  we  have  a  section  in  which  the  pitch  of  the  thread 
varies  from  top  to  bottom. 

The  interference  in  the  ordinary  type  of  worm-gear  is  absent 
from  the  Hindley  type,  and  the  consequent  undercutting  and 
weakening  of  the  teeth,  therefore,  is  a  feature  with  which  the 
designer  of  the  Hindley  worm  gearing  does  not  have  to  contend. 
For  this  reason  we  are  not  limited  in  the  length  of  teeth,  by  inter- 
ference, as  in  the  ordinary  case.  This  fact  permits  a  wide  lati- 
tude in  the  choice  of  tooth  shapes  and  proportions.  In  most 
examples  we  will  find  that  the  depth  of  thread  is  much  greater 
in  proportion  to  the  thickness  than  in  the  ordinary  worm-gear, 


WORM 


Machinery 


Fig.  2.     Section  of  Ordinary  Worm  and  Worm-wheel  on  Central  Plane 

in  which  the  height  is  limited  by  reason  of  the  interference  at 
the  top  and  root  of  the  teeth. 

Nature  of  Contact  of  Hindley  Worm  Gearing.  —  The  general 
idea  of  the  Hindley  worm  gearing  is  that  there  is  surface  contact 
between  the  worm  and  gear,  and  that  the  contact  is  generally 
over  the  whole  number  of  teeth  in  mesh.  If  such  were  the  actual 
conditions,  the  Hindley  type  would  surely  be  an  ideal  mechan- 
ism for  high  velocity  ratios,  but  that  such  is  not  the  fact  is  the 
purpose  of  this  treatise  to  point  out.  That  the  contact  is  of  a 
superior  nature  we  will  not  deny,  nor  that  it  is  much  nearer 
a  surface  contact  than  exists  in  the  ordinary  worm-gear.  As  a 
means  of  comparison  Figs.  2  and  3  are  shown.  Fig.  2  shows  an 
axial  section  taken  through  the  worm  and  gear  of  the  ordinary 


HINDLEY   WORM  AND   GEAR 


205 


type,  while  Fig.  3  shows  a  similar  section  through  the  Hindley 
worm  and  gear.  The  "airy"  appearance  of  Fig.  2  as  compared 
with  Fig.  3  indicates  a  vast  difference  in  the  nature  of  contact, 
and  gives  the  advantage  to  the  Hindley  type,  wherein  is  the  origin 
of  certain  false  ideas  in  favor  of  the  latter.  These  illustrations 
also  show  peculiar  differences  in  the  action  of  the  two  types. 
The  absence  of  rolling  action  in  Fig.  3  is  the  most  prominent, 
and  it  shows  the  similarity  between  this  type  of  gear  and  a  screw 
and  nut. 

From  an  inspection  of  Fig.  3  we  may  feel  sure  that  the  contact 
on  the  axial  plane  is  as  shown,  but  as  to  the  nature  of  contact 
in  a  plane  either  side  of  the  middle  plane  we  are  in  the  dark  so 


GEAR 


Machinery 


Fig.  3.     Section  of  Hindley  Worm  and  Gear  on  Central  Plane 

far  as  the  drawing  illustrates.  Mr.  George  P.  Grant  says  con- 
cerning the  contact  of  the  Hindley  worm  and  gear:  "It  is  com- 
monly but  erroneously  stated  that  the  worm  (Hindley)  fits  and 
fills  its  gear  on  the  axial  section.  .  .  .  It  has  even  been  stated 
that  the  contact  is  between  surfaces,  the  worm  filling  the  whole 
gear  tooth.  ...  It  is  also  certain  that  it  (the  contact)  is  on  the 
normal  and  not  on  the  axial  section,  and  that  the  Hindley  hob 
will  not  cut  a  tooth  that  will  fill  any  section  of  it.  The  contact 
may  be  linear  on  some  line  of  no  great  length,  but  it  is  probably 
a  point  contact  on  the  normal  section. " 

It  is  not  clear  what  reason  Mr.  Grant  had  for  saying  that  the 
contact  is  normal  instead  of  axial,  because  there  seems  to  be 
good  reason  to  believe  that  the  contact  is  on  the  axial  section 
since  it  is  on  this  section  that  the  teeth  of  the  hob  have  a  com- 


206  WORM   GEARING 

mon  pitch.  The  teeth  have  not  a  common  pitch  on  any  section 
at  an  angle  with  the  axial  section.  For  what  reason  would  one 
expect  to  find  contact  on  the  normal  section  in  this  case  any 
more  than  in  the  case  of  the  ordinary  worm?  Since  both  styles 
of  worm-wheels  are  hobbed  with  a  revolving  hob  which  lies  in  a 
plane  perpendicular  to  the  axis  of  the  worm-wheel,  the  contact 
could  hardly  be  on  a  normal  section. 

Professor  MacCord  states  that  he  considers  the  contact  to  be 
line  contact  on  the  axial  section,  and  he  gives  directions  for 
obtaining  the  exact  nature  of  the  contact  and  also  the  thread  and 
tooth  sections.  These  directions,  on  account  of  the  compli- 
cated nature  of  the  method,  are  difficult  to  follow.  Much, 
however,  can  be  found  out  by  simple  methods.  In  what  follows, 
describing  these  simpler  methods,  the  results,  of  course,  are  of  an 


HELIX  LINE  OF  HINDLEY  WORK 


HELIX  LINE  OF  ORDINARY  WOR 


Machinery 


Fig.  4.     Development  of  Ordinary  Worm  and  Hindi ey  Worm  Spirals 
on  a  Plane 

approximate  order,  but  they  nevertheless  give  a  means  of  com- 
parison and  a  material  basis  for  the  line  of  argument. 

The  Ideal  Case  Considered.  —  It  is  assumed  that  we  are  ex- 
amining an  ideal  Hindley  gear  in  which  the  worm  and  wheel  are 
theoretically  correct  in  shape  and  that  the  surfaces  are  perfectly 
smooth  and  inelastic.  From  the  nature  of  the  worm,  the  helix 
angle  varies  from  mid-section  to  the  ends,  decreasing  as  the 
thread  approaches  the  ends  of  the  worm.  The  thread  is  spiral 
as  well  as  helical.  This  change  in  the  thread  angle  is  caused  by 
the  increase  in  diameter  at  the  ends  of  the  worm  and  by  the  fact 
that  the  axial  pitch  of  the  thread  decreases  as  it  reaches  the 
ends.  The  decrease  in  axial  pitch  is  due,  of  course,  to  the  circular 
path  of  the  threading  tool.  If  we  take  a  development  on  a  flat 
surface  of  a  line  scribed  in  the  spiral  path  on  the  worm  blank, 
as  shown  in  Fig.  4,  the  change  in  the  angle  becomes  noticeable. 


HINDLEY  WORM  AND   GEAR 


207 


In  the  operation  of  forming  the  teeth  of  the  gear,  the  blank  is 
rotated,  each  portion  of  the  hob  working  the  tooth  into  shape 
so  that  it  will  pass  the  corresponding  portion  of  the  worm  thread 
without  interference,  permitting  a  smooth  transmission  of  mo- 
tion. If  each  portion  has  a  different  shape  or  is  placed  in  a  differ- 
ent relation,  the  shape  of  a  gear  tooth  will  be  a  compromise 
between  the  extremes,  and  this  is  what  is  actually  the  result,  as 
we  shall  see  later. 

The  progressive  steps  of  the  process  are  shown  in  Fig.  5 ;  the 
successive  positions  of  one  tooth  are  shown,  beginning  at  the 
left  and  ending  at  the  right-hand  position  where  each  tooth  is 


Machinery 


Fig.  5.    Successive  Steps  in  Shaping 
the  Hindley  Worm 


Fig.  6.    Surfaces  of  Contact  of  the 
Hindley  Worm 


given  its  final  shape.  The  nature  of  the  process  is  shown  in  Fig.  6, 
the  shaded  portions  representing  the  gear  teeth.  Here  we  have 
a  representation  of  the  contact  of  the  thread  and  teeth;  it  shows 
that  surface  contact  is  impossible  on  any  but  the  heavily  shaded 
portions  of  the  teeth,  it  being  confined  to  the  mid-section  and 
the  extreme  end  sections  of  the  worm.  Line  contact  is  obtained 
throughout  the  length  of  the  worm  on  the  axial  plane.  This 
figure  also  shows  that  no  advantage  is  gained  in  surface  contact 
by  making  the  worm  of  greater  length.  The  location  of  the  con- 
tacts are  shown  in  Fig.  6,  at  a,  s,  s,  s,  s,  a,  but  it  must  be  re- 
membered that  they  lie  on  opposite  sides  of  the  cutting  plane. 
From  this  it  is  apparent  that  the  worm  does  not  entirely  fill  the 


208 


WORM   GEARING 


space  between  the  teeth  of  the  gear  and  that  the  contact  is  not 
wholly  a  surface  contact. 

Let  us  investigate  still  further  and  see  whether  the  conditions 
are  not  modified  by  other  irregularities:  Fig.  7  is  drawn  to  repre- 
sent a  worm  and  gear  of  the  Hindley  type,  in  mesh,  the  teeth  of 
which  have  no  depth.  As  before  mentioned,  the  peculiarity  of 
this  type  of  worm  is  its  hour-glass  shape.  The  hob  and  worm 
may  be  treated  as  identical  in  form.  In  the  process  of  genera- 
tion, the  tooth  has  a  pitch  line  curvature  that  changes  with  corre- 
sponding positions  in  relation  to  the  thread  portion  acting  upon 


Machinery 


Fig.  7.     Effect  of  Hour-glass  Shape  on  Worm-wheel  Contact 

it.  The  tooth  must  necessarily  be  modified  from  what  it  should 
be  for  any  particular  location  in  its  contact  with  the  worm 
thread.  It  is  quite  clearly  shown  that  if  the  tooth  is  to  fill  the 
worm  thread  or  vice  versa,  it  must  be  formed  in  strict  accordance 
with  the  thread  at  that  particular  point.  Thus  if  at  j  the  tooth 
fills  the  thread,  that  tooth  must  be  formed  by  the  thread  at  that 
point,  while  the  tooth  at  k  must  be  formed  by  the  thread  at  k. 
Now,  since  each  tooth  must  pass  from  k  toj,  its  form  must  be 
such  that  it  will  do  so  without  interference.  It  is  evident  that 
the  radial  section  of  the  gear  at  k  must  be  the  same  as  at  j. 
Since  the  worm  is  largest  in  diameter  at  &,  the  curvature  of  the 
tooth  on  the  radial  section  is  dependent  on  the  thread  at  that 


HINDLEY  WORM  AND  GEAR  209 

point.  The  curvature  of  the  tooth  at  k  evidently  is  that  of  an 
ellipse  whose  major  axis  is  AA\.  Now,  since  the  thread  is  made 
with  angular  sides,  the  hob  could  hardly  act  on  the  teeth  of  the 
gear  the  same  at  all  points  from  k  to  j  except  on  the  axial  plane 
where  the  relative  shape  of  the  hob  thread  is  the  same  for  any 
position  along  the  line  of  action  (see  Fig.  3).  This  is  evident 
from  Fig.  7  at  E,  which  point  only  touches  at-, the  mid -section 
of  the  worm.  Therefore  we  still  have  the  line  contact  from 
top  to  bottom  of  teeth  on  the  axial  plane,  but  the  construction, 
Fig.  7,  shows  that  the  surface  contact  s,  -s,  s,  s,  Figs.  3  and  6, 
does  not  actually  exist,  but  that  the  surface  contact  at  the  ends 
of  the  worm  remains  undisturbed. 

From  the  above  we  may  safely  conclude  that  the  hob  at  j 
has  but  little  effect  on  the  actual  shape  of  the  tooth,  and  that  its 
influence  increases  until  k  is  reached.  Fig.  7  also  shows  a  good 
reason  why  the  contact  may  be  considered  axial  instead  of  nor- 
mal, by  the  mere  fact  of  the  differences  in  curvature  of  worm  and 
wheel  at  any  point  other  than  k.  In  practice  the  contact  may 
appear  to  be  surface  contact,  but  this,  no  doubt,  is  due  to  the 
influence  of  the  lubricating  oil  and  the  fact  that  materials  of 
construction  are  distorted  to  some  extent  in  form  when  sub- 
jected to  pressure.  This  distortion  permits  the  worm  thread  to 
imbed  itself  into  the  worm-wheel  teeth,  somewhat  broadening 
the  contact  for  the  time  being.  The  conditions  as  stated  in  the 
above  discussion  would  be  met  in  the  case  of  a  hardened  worm 
and  gear  with  surfaces  finished  by  lapping.  In  practice  the 
worm  and  gear  are  ground  together,  sand  and  water  being  used 
as  the  abrasive.  This  grinding  wears  down  the  roughness  of  the 
surfaces  and  tends  to  correct  irregularities  in  form  that  develop 
in  the  hobbing  process. 

Objections  to  the  Hindley  Gear.  —  The  objections  to  the 
Hindley  type  of  worm-gear  are  many  and  are  widely  known. 
It  must  be  set  up  accurately,  the  alignment  being  made  perfect. 
End  play  is  a  feature  that  must  be  avoided,  as  any  longitudinal 
displacement  of  the  worm  will  cause  the  gear  to  cut.  These 
peculiarities  are  the  greatest  drawbacks  to  the  use  of  this  gear, 
and  because  of  them  the  author  believes  that  it  will  not  come  into 


2IO 


WORM   GEARING 


common  use,  at  least  not  so  common  as  the  worm  drive  of  the 
ordinary  type.  This  opinion  is  strengthened  by  the  fact  that 
we  have  become  so  much  more  familiar  with  the  latter  type  as 
to  be  able  to  design  and  construct  drives  that  work  satisfactorily 
in  every  respect. 

Modifications  of  Hindley  Worm-gear  Practice.  —  Some  mod- 
ifications have  been  made  in  the  process  of  manufacturing  the 
Hindley  worm-gear.  One  that  is  probably  of  first  importance 
is  that  known  as  the  "  second  cut. "  The  effect  of  the  second  cut 
is  indicated  in  Fig.  8.  From  this  illustration  one  would  say  that 
the  object  of  the  second  cut  is  to  remove  the  points  of  contact. 
Whether  this  is  the  reason  or  not,  it  is  a  fact  that  it  does  remove 


WORM 


Machinery 


Fig.  8.     Effect  of  the  "Second  Cut"  on  Contact 

considerable  of  the  contact  from  all  but  the  mid-section  of  the 
worm.  This  second  cut  is  made  by  enlarging  the  diameter  of 
the  circle  in  which  the  threading  tool  travels  when  cutting  the 
worm.  It  is  said  to  have  advantages  that  add  to  the  wearing 
quality  of  the  drive,  but  just  what  these  advantages  are  is  not 
apparent,  and  since  the  process  is  considered  more  or  less  a  trade 
secret,  it  is  difficult  to  obtain  authentic  reasons  for  its  use. 

The  limiting  length  of  the  worm  is  dependent  on  the  shape  of 
the  thread.  In  Fig.  8  the  worm  is  shown  with  three  teeth  in 
mesh,  while  Fig.  3  shows  five.  Fig.  3  shows  a  case  that  would 
be  impossible  in  practice  on  account  of  the  undercut  teeth 
A  which  lock  the  worm  in  mesh.  The  side  of  the  thread  must 


HINDLEY  WORM  AND   GEAR  211 

fall  inside  the  line  be  to  permit  the  worm  and  gear  to  be  as- 
sembled. 

Conclusions  Regarding  the  Hindley  Worm  and  Gear.  —  The 
following  are  the  conclusions,  derived  from  the  investigation 
regarding  the  Hindley  type  of  gear: 

1.  The  contact  is  purely  sliding  contact. 

2.  The  nature  of  the  contact  is  linear,  closely  resembling  sur- 
face contact. 

3.  Linear  contact  extends  from  the  top  to  the  root  of  the 
tooth. 

4.  The  contact  is  on  the  axial  section. 

5.  The  thread  section  fills  the  tooth  space  on  the  axial  section 
only. 

6.  The  mid-portion  of  the  hob  has  little  or  no  effect  in  shaping 
the  teeth  of  the  gear. 

7.  Surface  contact  exists  on  opposite  sides  of  the  axial  plane 
at  the  end  of  the  worm  thread  and  is  intermittent  in  nature, 
because  the  end  of  the  thread  passes  out  of  contact  with  the 
tooth  in  the  revolving  of  the  worm.     This  contact  is  on  a  plane 
normal  with  the  thread  angle. 

In  practice  it  is  usual  to  allow  considerable  back-lash  between 
the  thread  and  the  tooth  of  the  worm  and  gear.  This  play  tends 
to  counteract  bad  workmanship,  either  in  construction  or  erec- 
tion. 


CHAPTER  X 
METHODS  FOR  FORMING  THE  TEETH  OF  WORM-WHEELS 

To  correctly  classify  and  comprehend  the  various  methods  and 
machines  for  cutting  the  teeth  of  worm-wheels,  it  is  first  neces- 
sary to  clearly  define  the  term  "worm  gearing."  We  will  con- 
sider that  by  worm  gearing  we  mean  gearing  of  the  type  of  which 
a  cross-section  is  shown  at  the  left  of  Fig.  i,  in  which  the  acting 
face  of  the  wheel  is  curved  to  fit  the  form  of  the  worm,  and  in 
which  the  whole  width  of  the  wheel  face  is  in  active  working  con- 
tact with  the  worm. 

The  action  is  best  understood  by  taking  vertical  sections  on 
the  center  line  A- A,  and  other  lines  such  as  that  at  B-B,  parallel 
with  the  center  line.  Sections  on  lines  A-A  and  B-B  are  shown 
at  the  right  of  the  cut.  With  worm  gearing  of  standard  form, 
the  section  on  line  A-A  shows  the  worm  to  have  the  profile  of 
an  involute  rack,  while  the  teeth  of  the  wheel  show  outlines 
identical  with  those  of  the  corresponding  involute  gear  of  the 
same  pitch  and  number  of  teeth,  suited  to  engage  with  the  rack. 
In  other  words,  the  teeth  of  the  gear  are  such  as  would  be  formed 
by  the  teeth  of  the  worm  if  the  latter  acted  as  a  rack  in  a  mold- 
ing-generating operation.  A  section  on  line  B-B  shows  that  the 
teeth  of  the  worm  have  a  distorted  outline  on  planes  removed 
from  the  axial  plane.  If  we  consider  these  distorted  teeth  as 
the  teeth  of  a  rack,  molding  their  mating  tooth  spaces  in  a  gear 
running  on  the  same  center  as  the  worm  gear  and  at  the  same 
speed,  it  will  form  the  distorted  wheel  teeth  shown  for  the  section 
on  line  B-B.  In  a  word,  each  section  of  the  worm  parallel  to 
the  axial  section  A-A  is  a  rack  section,  which  molds  in  the  wheel 
below  it  the  proper  teeth  to  mesh  with  it  in  accurate  conjugate 
action.  The  true  worm-wheel,  it  is  thus  seen,  must  be  formed 
by  the  molding-generating  process. 

The  same  worm  as  that  shown  in  Fig.  i  may  be  made  to  en- 

212 


METHODS  OF  CUTTING  TEETH 


2I3 


gage  with  a  spiral  gear  of  the  same  number  of  teeth  as  the  worm- 
wheel,  provided  the  teeth  are  of  the  proper  pitch  and  set  at  an 
angle  to  agree  with  the  helix  angle  of  the  worm.  The  action  of 
such  gearing,  however,  does  not,  like  that  in  Fig.  i,  take  place 
on  all  sections  A-A,  B-B,  etc.,  but  is  confined  to  a  point  at  or 
near  the  center  line  A- A.  The  contact,  in  other  words,  is  point 
contact,  and  not  line  contact  extending  clear  across  the  face  of 
the  wheel.  Such  a  combination,  in  fact,  is  not  a  case  of  worm 
gearing,  but  a  case  of  spiral  gearing  —  and  a  very  poor  case  at 
that. 


SECTION  ON    LINE  B-B 


Machinery 


Fig.  i.    Action  of  a  True  Worm-wheel 

Gashing  Worm-wheels  by  the  Formed  Cutter  Process.  — 

While  the  method  of  forming  a  true  worm-wheel  is  thus  seen  to 
be  accurately  performed  only  by  the  molding-generating  process, 
the  accurate  teeth  produced  by  that  process  may  be  closely 
approximated  in  many  cases  by  the  "gashing"  method,  which 
belongs  in  the  formed  cutter  classification.  In  this  operation  a 
milling  cutter  is  used  having  approximately  the  outline  of  a 
normal  section  of  the  teeth  of  the  worm  to  be  used.  This  cutter 
is  of  the  same  diameter  as  the  worm,  and  is  set  with  relation  to 
the  axis  of  the  work  at  the  helix  angle  of  the  worm,  as  measured 
on  the  pitch  line.  It  is  centered  over  the  wheel,  and  fed  into  the 
latter  to  the  proper  depth  to  form  a  tooth  space;  it  is  then  drawn 
out  again,  the  work  is  indexed  to  the  next  tooth  space,  and  the 


214  WORM  GEARING 

cutter  again  sunk  in  to  depth,  the  operation  being  repeated  until 
the  wheel  is  completed. 

A  universal  milling  machine  is  generally  used  for  this  oper- 
ation. With  the  table  set  at  90  degrees,  the  cutter  is  first  brought 
centrally  over  the  work  arbor  by  adjusting  the  saddle  on  the 
knee  of  the  milling  machine,  and  then  the  work  is  brought 
centrally  with  the  cutter  arbor  by  adjusting  the  table  by  the 
feed-screw.  The  work  table  is  next  swung  to  the  helix  angle  of 
the  worm  which  is  to  be  used  with  the  wheel.  Then  the  cutting 
is  proceeded  with. 

This  gashing  process  gives  a  tooth  very  closely  approximating 
the  true  tooth  form,  when  the  diameter  of  the  worm  is  large  as 
compared  with  the  pitch,  and  when  the  worm  is  single-threaded, 
but,  for  multiple-threaded  worms  of  smaller  diameter  in  propor- 
tion to  their  pitch,  the  process  is  impracticable.  This  method 
is,  however,  used  by  at  least  one  of  the  best-known  builders  of 
gear-cutting  machines  in  forming  the  teeth  in  the  index  worm- 
wheel.  It  is  used  under  the  conditions  which  give  a  very  close 
approximation  to  the  true  form  of  tooth,  and  is  employed  in  this 
particular  case  for  the  sake  of  the  high  degree  of  accuracy  obtain- 
able. The  index  wheel  is  divided,  in  cutting,  by  a  carefully- 
made  and  carefully-preserved  master  wheel.  The  step-by-step 
gashing  process  allows  the  spacings  of  this  superior  master  wheel 
to  be  accurately  reproduced  in  the  index  wheel  being  cut  —  more 
accurately,  it  is  claimed,  than  would  be  possible  if  it  were  to  be 
reproduced  by  the  nobbing  process. 

The  gashing  process  is  also  used  for  roughing  out  worm-wheels 
preparatory  to  hobbing.  In  a  previously  gashed  wheel,  as  will 
be  explained  later,  the  hobbing  operation  is  one  of  extreme 
simplicity,  not  requiring  special  machines  or  mechanisms  of  any 
kind. 

The  Molding-generating  Principle.  —  As  already  explained, 
the  molding-generating  principle  is  the  only  one  that  will  accu- 
rately form  the  teeth  of  worm-wheels.  The  principle  involved  is 
shown  in  Fig.  2.  The  forming  worm  (or  hob)  is  connected  by 
gearing  with  the  worm-wheel  blank  to  be  formed,  in  the  same 
ratio  as  in  the  finished  worm  gearing.  While  the  blank  and  the 


METHODS  OF  CUTTING  TEETH 


215 


forming  worm  are  rotated  together  in  this  ratio,  the  latter  is  fed 
into  the  blank  slowly,  its  threads  forming  the  properly  shaped 
tooth  in  the  wheel.  As  the  worm  revolves,  an  axial  section 
would  give  the  appearance  of  a  rack  like  that  shown  in  section 
A-A,  of  Fig.,  i,  moving  continuously  and  forming  suitable  gear 
teeth  in  the  wheel  below  it.  Any  other  section,  such  as  B-B  in 
Fig.  i,  would  also  act  as  a  distorted  rack,  forming  correspondingly 
distorted  gear  teeth  in  that  portion  of  the  worm-wheel  in  the 
same  plane. 


DRIVING  PULLEY 


Machinery 


Fig.  2.     Diagram  showing  the  Principle  of  the  Robbing  Process  for 
Cutting  Teeth  in  Worm-wheels 

Of  the  various  methods  of  operation,  by  which  the  molding- 
generating  principle  can  be  applied,  shaping  or  planing  is,  of 
course,  impracticable.  Milling  is  the  method  generally  em- 
ployed. Grinding  or  abrasion  is  used  to  a  limited  extent,  it 
being  sometimes  employed  in  the  case  of  "grinding  in"  a  worm 
with  a  wheel  already  roughly  cut  to  shape.  In  this  operation 
the  worm  and  wheel  are  run  together  in  place,  under  considerable 
pressure,  the  teeth  of  the  gear  being  liberally  supplied  with  oil 
and  emery,  which  act  as  an  abrasive  and  form  the  teeth  of  the 
gear  and  worm  to  fit  each  other. 


2l6  WORM   GEARING 

In  the  commonly  employed  milling  operation,  the  process  is 
that  known  as  "hobbing,"  and  the  milling  cutter  or  tool  used  is 
a  "hob."  The  hob,  barring  modifications  required  for  relief 
or  clearance,  and  allowance  for  regrinding,  as  explained  in  a 
following  chapter,  is  practically  a  replica  of  the  worm  which  is 
to  be  used,  but  with  grooves  cut  in  it  so  as  to  form  teeth.  This 
hob  is  rotated  in  the  proper  ratio  with  the  work,  exactly  as  shown 
in  Fig.  2,  and  fed  slowly  down  into  it,  cutting  out  the  tooth  spaces 
in  the  wheel  as  it  does  so.  When  it  has  reached  the  proper  depth, 
the  teeth  are  all  formed  to  the  proper  shape. 

Robbing  Worm-wheels  in  the  Milling  Machine.  —  The  sim- 
plest method  of  rotating  the  hob  and  the  work  in  the  proper 
ratio  with  each  other  is  that  in  which  the  work  is  first  gashed, 
and  then  finished  with  the  hob  in  such  a  way  as  to  be  driven 
by  the  latter,  the  work  and  the  hob  thus  furnishing  their  own 
driving  mechanism.  The  worm-wheel  is  mounted  so  as  to  re- 
volve freely  on  dead  centers.  This  is  the  simplest  method  of 
making  correct  worm-wheel  teeth.  It  does  not  require  special 
appliances  of  any  kind,  being  done  in  an  ordinary  milling  machine 
with  a  gashing  cutter  and  a  hob.  Complete  details  of  the  prac- 
tical operations  for  producing  worm-wheels  by  gashing  and  hob- 
bing will  be  given  in  the  following  chapter. 

In  cases  where  it  is  desired  to  hob  worm-wheels  directly  from 
the  solid  without  preliminary  gashing,  it  is  necessary  to  provide 
some  special  device  for  rotating  the  hob  and  the  work  in  unison 
as  in  Fig.  2.  Special  worm-wheel  hobbing  machines  are  made 
for  this  purpose.  One  of  the  questions  met  with  in  this  con- 
nection is  the  figuring  of  the  gearing  to  properly  connect  the 
hob  and  the  gear.  This  will  be  explained  in  the  following  para- 
graphs. 

Gearing  for  Worm-wheel  Hobbing  Machines.  —  The  manner 
in  which  the  machine  is  geared  will  depend  on  the  assortment 
of  change  gears  with  which  it  is  provided.  If  there  is  a  sufficient 
variety  of  these,  simple  gearing  may  be  employed,  using  an  idler 
on  the  swinging  arm  to  connect  the  gear  on  shaft  D  with  .gear  A, 

Fig.  3- 

First  find  the  revolutions  of  the  hob  for  each  revolution  of  the 


METHODS  OF   CUTTING  TEETH 


217 


worm-wheel.  This  is  found  by  dividing  the  number  of  teeth 
in  the  wheel  by  the  number  of  threads  in  the  hob,  or  worm, 
with  which  it  is  to  run.  For  instance,  if  there  are  fifty  teeth  in 
the  wheel  and  the  worm  is  single  threaded,  the  number  of  rev- 
olutions of  the  work  to  one  of  the  hob  will  be  50  -r-  i  =  50. 


DRIVING  SHAFT. 


MITER  GEARS 


. WORK  SPINDLE 


ELEVATION 


Fig.  3.     Arrangement  of  Gearing  in  Robbing  Machine 

This  may  be  called  the  ratio  of  the  wheel.  If  the  worm  is  double 
threaded,  the  ratio  will  be  50  -f-  2  =  25,  and  so  on.  With  the 
gear  connections  indicated  in  Fig.  3,  for  simple  gearing,  when  A 
has  96  teeth,  the  gear  on  shaft  D  must  have  a  number  of  teeth 
equal  to  4  times  the  ratio;  that  is  to  say,  if  we  have  a  worm-wheel 
with  50  teeth,  driven  by  a  double-threaded  worm,  the  ratio  is 


218 


WORM   GEARING 


25,  and  the  number  of  teeth  in  the  gear  at  D  =  4  X  25  =  100. 
If  a  48-tooth  gear  is  used  at  A  in  place  of  the  96-tooth  gear, 
the  gear  on  D  is  found  by  multiplying  the  ratio  by  2.  If,  for 
instance,  the  number  of  teeth  in  the  wheel  to  be  cut  is  135,  to 
mesh  with  a  triple-threaded  worm,  the  ratio  will  be  135  -5-3  =45 
and  the  number  of  teeth  for  gear  D  will  be  2  X  45  =  90,  when  A 
has  48  teeth. 

A  wider  range  of  ratios  can  be  provided  for  with  a  given  num- 
ber of  change  gears  if  A  and  D  are  connected  by  compound  gear- 


Machinery 


Fig.  4.     Compound  Gearing  for  Machine  shown  in  Fig.  3 

ing,  as  shown  in  Fig.  4,  where  A  and  C  are  the  driving  gears  and 
B  and  D  the  driven  gears.     The  rule  for  rinding  the  number  of 
teeth  for  B,  C  and  D  when  A  has  96  teeth  then  becomes: 
number  of  teeth  m  B  X  number  of  teeth  in  D 


number  of  teeth  in  C 
When  A  equals  48,  this  becomes: 
number  of  teeth  in  B  X  number  of  teeth  in  D 
number  of  teeth  in  C 


=  4  X  ratio. 


=  2  X  ratio. 


Suppose,  for  instance,  that  we  have  a  set  of  change  gears 
varying  by  6,  that  is  to  say,  the  numbers  run  18,  24,  30,  36,  42, 
etc.,  from  18  to  120.  Suppose  the  ratio  of  the  worm-gear  to  be 
cut  is  50,  and  the  number  of  teeth  in  gear  A  is  96;  then  we  have: 
number  of  teeth  in  gear  B  X  number  of  teeth  in  gear  D 
number  of  teeth  in  gear  C 

=  4  X  50  =  200. 


METHODS  OF  CUTTING  TEETH  219 

By  selecting  a  6o-tooth  gear  for  B,  a  i2o-tooth  gear  for  D  and 
a  3  6-  tooth  gear  for  C,  we  have: 

60  X  120 
-^--4X50. 

Proving  the  calculations  of  the  gearing  for  this  machine  is 
practically  the  same,  whether  simple  or  compound  gearing  is 
used.  Consider  that  the  whole  mechanism  is  driven  from  the 
hob  spindle;  then  the  product  of  all  the  driven  gears,  divided  by 
the  product  of  all  the  driving  gears,  equals  the  number  of  teeth 
in  the  worm-wheel  divided  by  the  number  of  threads  in  the 
worm  or  hob.  An  idler  gear  between  a  driving  and  driven  gear 
is  not  considered  at  all,  as  it  has  no  effect  on  the  motion  other 
than  to  reverse  it.  Proving  the  first  example  by  this  method, 
we  have  : 


. 

I  I          96          96          2 

The  number  of  teeth  in  these  driving  and  driven  gears  are 
given  in  their  order  from  the  work,  through  the  mechanism,  to 

the  hob.     The  fraction  -  represents  the  miter  gears,  which  are 

of  even  ratio,  but  the  number  of  teeth  of  which  are  not  given. 
For  the  last  example  the  proof  is  similar.     Here  we  have: 


. 

I  I  36  96  96  I 

The  Fly-tool  Method  of  Cutting  Worm-wheels.  —  By  pro- 
viding suitable  driving  and  feeding  mechanism,  it  is  possible  to 
use  a  simple  fly-cutter  for  forming  the  teeth  of  worm-wheels 
in  place  of  the  expensive  hob  used  in  the  operations  previously 
described.  The  movements  required  for  this  method  will  be 
understood  from  a  study  of  Fig.  5.  Here  is  shown  in  dotted 
lines  a  worm  meshing  with  a  worm-wheel,  a  portion  only  of  the 
periphery  of  which  is  seen.  Such  a  worm,  properly  located  with 
reference  to  a  plastic  blank  and  rotating  with  it  in  the  proper 
ratio,  will  form  accurate  teeth  in  the  latter  by  the  molding- 
generating  process.  Gashing  this  worm  makes  of  it  a  cutter  by 
means  of  which  the  same  form  may  be  given  to  a  blank  of  solid 


220 


WORM   GEARING 


metal.     The  teeth  of  such  a  gashed  hob  coincide  with  the  out- 
lines of  the  thread  of  the  worm. 

In  Fig.  5,  in  full  lines,  is  shown  a  cutter  bar  with  a  blade  T\ 
of  the  same  outline  as  the  thread  of  the  worm  and  the  tooth  of 
the  corresponding  hob.  In  order  to  permit  this  single  cutting 
tool  to  perform  the  function  of  the  worm  as  it  molds  the  plastic 
substance,  or  of  the  hob  as  it  cuts  its  shape  in  the  metal,  it  must 
be  fed  helically  as  the  bar  and  work  revolve,  following  the  out- 
lines of  the  imaginary  worm  from  one  end  to  the  other  as  the 
cutting  progresses.  Beginning  at  the  left,  for  instance,  the 


IMAGINARY  WORM,   FORMING  TEETH   IN  THE  WHEEL  BY  THE  MOLDING-GENERATING  PROCESS. 

CUTTER  BAR  WITH  BLADE,  T^ ,  WHICH  PROGRESSIVELY  FOLLOWS  THE\ 
THREAD  OF  THE   WORM,  AND  SO  CUTS  THE  TEETH  OF  THE  WHEEL.     J\ 


/  /I      '  i'    fit  /'  '/  /  /(W  '  // 

/  /  //  //  r  n  V  luiuL 

/H//II/I  //f/ffi 


Machinery 


Fig.  5.     Diagram  showing  the  Principle  of  the  Fly-tool  Method  of 
Cutting  the  Teeth  of  Worm-wheels 

blade  may  be  fed  helically  in  the  line  of  the  thread,  passing 
through  positions  7\  and  T3,  until  the  feed  finally  runs  out  at 
the  extreme  right. 

The  methods  of  giving  this  progressive  helical  change  of  posi- 
tion to  the  fly-cutter  are  various.  It  would  be  possible,  for 
instance,  to  so  connect  the  feed-screw,  by  which  the  cutter-bar  is 
advanced  with  the  rotating  mechanism  for  the  bar,  through 
differential  and  change  gearing,  that  a  rotating  movement  due 
to  the  axial  feeding  of  the  latter  would  be  added  to  or  imposed 
upon  the  rotation  due  to  its  connection  with  the  work,  just  as, 


METHODS  OF  CUTTING  TEETH  221 

in  Fig.  n,  Chapter  IV,  the  rotation  due  to  the  downward  feed 
of  the  cutter  slide  is  combined  with  that  due  to  the  connection 
with  the  cutter  spindle  for  rotating  the  work.  If  the  proper 
change  gears  were  selected  so  that,  with  the  spindle-  and  work- 
driving  mechanisms  stationary,  the  feeding  forward  of  the  cutter 
bar  would  rotate  the  latter  at  the  proper  rate  to  give  the  lead  of 
the  work,  the  blade  would  evidently  follow  the  path  of  the  thread 
of  the  imaginary  worm,  as  shown  at  TI  and  T3  in  Fig.  5.  Owing 
to  the  action  of  the  differential  mechanism,  it  would  still  follow 
the  thread  of  the  imaginary  worm,  even  if  the  latter,  with  the 
spindle-  and  work-driving  mechanisms,  were  in  motion. 

Another  method  consists  in  combining  in  the  work,  also  by 
differential  gearing,  a  rotation  due  to  the  revolving  of  the  cutter 
with  a  rotation  due  to  the  axial  feed  of  the  cutter-bar.  That 
this  produces  the  same  effect  as  the  previous  arrangement  will 
also  be  understood  from  Fig.  5. 

First,  let  the  rotation  of  the  cutter  be  arrested.  If  the  cutter- 
bar  with  a  worm  mounted  on  it,  such  as  shown  by  the  dotted 
lines,  be  now  fed  axially  in  the  direction  of  the  arrow,  the  positive 
connections  between  the  feed  and  the  work  spindle,  through  the 
change  gearing  and  the  differential  gearing,  will  cause  the  work 
to  rotate  uniformly  with  it.  If  the  feed  is  arrested  after  a  time, 
and  the  bar  is  started  revolving,  the  imaginary  worm  mounted 
on  it  will  still  be  kept  in  proper  mesh  with  the  work,  owing  to 
the  change  gear  connections  between  the  cutter-bar  and  the 
work  spindle,  acting  through  the  differential  gearing.  As  pre- 
viously explained,  the  office  of  the  differential  gearing  is  to  com- 
bine in  the  work  the  rotation  due  to  the  feeding  and  that  due  to 
the  rotation  of  the  worm,  in  such  a  way  that  they  can  take  place 
simultaneously  as  well  as  separately;  so  that  it  will  be  seen  that 
if  the  connections  are  properly  made,  the  worm  may  be  fed  end- 
wise and  revolved  at  the  same  time,  always  keeping  in  perfect 
step  with  the  work. 

Now,  the  imaginary  worm  and"  the  fly-tool  are  both  firmly 
fixed  to  the  cutter-bar,  so  that  the  fly-tool  must  always  follow 
the  movements  of  the  imaginary  worm.  Being  set  to  coincide 
with  the  outlines  of  the  worm  thread  at  the  start,  it  must  always 


222 


WORM   GEARING 


coincide  with  those  outlines,  and  since  the  worm  is  never  out  of 
step  with  the  work,  the  fly-tool  will  never  be  out  of  step  either. 
It  will  thus  be  seen  that  it  will  always  follow  the  helical  path 
of  the  dotted  lines  in  Fig.  5,  in  moving,  for  instance,  from  7\  to 
T3.  Revolving  in  the  position  T3,  T4,  T2,  etc.,  the  work,  as 
shown  in  the  dotted  lines  of  T2,  will  always  be  in  proper  relation 
with  the  fly-tool,  as  it  is  with  the  imaginary  worm. 


STARTING  POSITION  OF  HOB 


m- 

POSITION  OF  HOB  AT  END  OF  CUT 

Machinery 


Fig.  6.     Diagram  Illustrating  Manner  in  which  a  Tapered  Hob  is 
Presented  to  a  Worm-wheel  Blank 

With  this  arrangement,  if  the  change  gearing  connecting  the 
driving  mechanism  of  the  cutter-bar  and  the  work  were  dis- 
connected while  the  bar  were  fed  through  from  left  to  right,  the 
rotary  motion  given  by  the"  connection  of  the  feed  of  the  bar 
with  the  work  would  shape  one  tooth.  If,  on  the  other  hand, 
the  gearing  connecting  the  feed  of  the  bar  with  the  rotation  of 
the  work  were  disconnected  while  the  connections  between  the 


METHODS   OF  CUTTING  TEETH 


223 


drive  of  the  bar  and  the  work  were  in  operation,  the  cutter  would 
partially  shape  each  tooth  of  the  work.  By  combining  the  two 
movements  in  the  differential  gearing,  the  cutter  perfectly  forms 
all  the  teeth. 

Tapered  Hob  for  Cutting  Worm-wheels.  —  When  cutting 
worm-wheels  by  this  method,  the  hob,  as  indicated  in  Fig.  6, 
is  tapered.  It  is  placed  on  the  cutter  spindle  and  fed  axially 


INDEX  WORM  TRAVERSING  SCREW 


Machinery 


Fig.  7. 


Diagram  of  Original  Form  of  Mechanism  for  Generating 
Worm-wheels  with  Taper  Hob 


past  the  work  in  the  same  way  that  the  fly-tools  in  the  previous 
case  are  fed.  The  combined  movements  cause  the  hob  to  follow 
spirally  in  the  path  of  the  thread  of  the  imaginary  rotating  worm. 
The  small  end  of  the  hob  first  commences  to  work,  and  as  the 
cutter  spindle  is  fed  forward,  the  cut  is  taken  successively  on 
larger  and  larger  diameters,  until  finally,  when  the  tool  has 
passed  clear  through,  the  full-sized  teeth  at  the  rear  end  of  the 


224  WORM   GEARING 

hob  complete  the  work.  The  machine  used  is  practically  iden- 
tical with  that  shown  in  Fig.  6,  Chapter  IV,  it  being  adapted  to 
cutting  worms  by  the  same  process.  The  differential  mechanism 
used  is  the  same  as  in  the  illustration  referred  to,  the  axial  feed 
of  the  cutter  spindle  being  applied  to  shaft  M,  while  the  rotative 
movement  of  the  cutter  spindle  is  connected  with  shaft  H,  the 
two  being  combined  in  gears  /,  L  and  N  to  rotate  the  indexing 
wheel  G. 

The  original  machine  for  this  purpose,  built  by  Mr.  Reinecker, 
Chemnitz,  Germany,  employed  a  different  form  of  combining 
or  differential  movement.  It  is  shown  diagrammatically  in 
Fig.  7.  In  this  case  the  tapered  hob  is  connected  by  change 
gearing  with  the  worm  driving  the  indexing  wheel,  as  before. 
The  worm,  however,  is  mounted  on  a  slide,  allowing  it  a  con- 
siderable range  of  axial  movement.  This  axial  movement  is 
controlled  by  a  screw  and  nut,  as  shown.  This  screw  is  con- 
nected by  change  gearing  with  the  screw  by  which  the  taper 
hob  is  fed.  It  will  thus  be  seen  that  the  feeding  of  the  hob 
rotates  the  work  by  shifting  the  index  worm  lengthwise,  while 
the  rotating  of  the  hob  rotates  the  work  through  the  rotation  of 
the  index  worm  and  worm-gear.  The  two  movements  are  inde- 
pendent of  each  other,  but  are  combined  with  the  same  effect 
as  produced  by  the  " jack-in-the-box"  differential  gearing  pre- 
viously described.  With  this  arrangement  the  ratio  of  table 
movement  and  lengthwise  worm  movement  should  be  propor- 
tioned in  the  ratio  of  the  pitch  diameters  of  the  worm-wheel 
being  cut,  and  the  index  worm-wheel.  The  reason  for  abandon- 
ing this  construction  was  doubtless  its  limited  range  of  move- 
ment, which,  though  sufficient  for  the  hobbing  of  worm-wheels, 
was  not  sufficient  (when  applied  to  the  universal  gear-cutting 
machine)  for  cutting  spiral  pinions  of  great  helix  angle. 

Ordinary  and  Taper  Hob  Method  of  Robbing  Worm-wheels 
Compared.  —  By  applying  the  hob  to  the  work  in  the  method 
just  described,  a  theoretically  correct  and  accurate  worm-gear 
is  produced,  which  cannot  be  excelled  where  high  pitch  angles 
or  wide  wheel  faces  —  wide  in  relation  to  the  worm  —  are  con- 
cerned. By  such  a  method  of  production  the  area  of  contact 


METHODS  OF   CUTTING  TEETH 


225 


between  the  worm  and  worm-wheel  is  as  large  as  possible,  and 
provided  that  a  good  combination  of  materials  is  used,  viz.,  a 
high  grade  of  phosphor-bronze  for  the  worm-wheel  and  a  good 
grade  of  casehardening  steel  for  the  worm,  this  type  of  gearing 
shows  no  appreciable  wear  under  the  heaviest  loads  after  having 
run  for  a  long  time.  In  cutting  the  worm-wheel  in  this  manner, 
the  reverse  curve  AB,  Fig.  8,  is  actually  obtained  in  the  worm- 
wheel  tooth.  This  curve  conforms  to  the  path  followed  by  a 
tooth  of  the  worm  in  rotating,  and  in  practice  is  considered  to 
give  a  surface  contact  which  is  impossible  to  obtain  by  hobbing 
worm-wheels  in  the  ordinary  way. 

The  method  ordinarily  used  in  hobbing  worm-wheels  is  to 
gear  up  the  mechanism  driving  the  hob  with  that  actuating  the 


Machinery 


Fig.  8.  Diagram  Illustrating  Reverse  Curve  that  is  produced  on 
Worm-wheel  Teeth  by  Hobbing  with  a  Tapered  Hob  fed  Longitu- 
dinally and  at  Right  Angles  to  the  Axis  of  the  Worm-wheel  Blank 

dividing  wheel  which  controls  the  indexing  or  rotation  of  the 
worm-wheel,  and  to  feed  the  hob  radially  into  the  worm-wheel, 
both  hob  and  blank  being  rotated  at  the  same  time.  This  ac- 
tion is  continued  until  the  hob  has  been  fed  in  to  the  correct 
depth.  Now  in  analyzing  this  method  of  cutting  the  worm-wheel, 
it  will  be  seen  that  by  presenting  the  hob  in  this  manner  it 
does  not  produce  a  tooth  of  a  theoretically  correct  shape,  for 
the  simple  reason  that  the  hob  cuts  away  certain  portions  of 
the  tooth  that  are  necessary  to  give  a  perfect  contact  with  the 
worm.  This  is  due  to  the  constant  changing  of  the  theoretical 
helix  angle  of  the  hob  while  being  fed  in  axially. 

Instead,  therefore,  of  producing  a  worm-wheel  tooth  that  has 
a  reverse  curve  corresponding  with  the  path  through  which 


226  WORM   GEARING 

the  face  of  the  worm  tooth  travels  in  rotating,  this  method  re- 
moves a  certain  amount  of  the  surface  of  the  worm-wheel  tooth 
that  should  come  in  contact  with  the  worm,  and,  in  theory,  the 
contact  between  a  worm-wheel  (cut  in  this  manner)  and  the 
worm  is  only  at  the  center.  The  worm-wheel  teeth  are  relieved 
toward-  each  end  and  are  not  in  contact  with  the  teeth  of  the 
worm,  these  portions  of  the  worm-wheel  teeth  being  removed  by 
the  hob  in  forming  them. 

A  very  high  degree  of  accuracy  can  be  obtained  by  the  taper 
hob  method  of  hobbing  worm-wheels,  owing  to  the  fact  that 
the  full  size  of  the  hob  does  not  come  into  play  until  the  finishing 
cut  is  reached,  so  that  the  teeth  of  the  hob  tend  to  preserve  their 
shape  indefinitely.  Another  point  that  tends  to  produce  accu- 
racy is  the  fact  that  the  distance  between  the  work  arbor  and  the 
hob  spindle  is  at  all  times  fixed  at  exactly  the  distance  between 
the  axis  of  the  worm-wheel  and  the  worm  in  the  finished  gearing. 
This  is  a  refinement  of  greater  importance  than  is  usually  realized, 
and  one  that  is  not  always  looked  out  for  in  hobbing  operations 
in  which  the  cutter  spindle  is  fed  in  toward  the  work. 

Efficiency  of  Taper-hobbed  Worm  Gearing.  —  In  order  to 
prove  that  worm-wheels  cut  by  this  method  would  work  out  as 
satisfactorily  in  practice  as  theoretical  considerations  indicated, 
a  number  of  tests  were  made  by  a  prominent  concern  manufac- 
turing pleasure  electric  cars.  In  these  tests  it  was  found  that 
the  efficiency  was  very  high,  averaging  from  90  to  98  per  cent. 
Continual  running  appeared  to  have  but  very  little  effect  on  the 
efficiency,  and  the  wear  was  almost  negligible;  the  only  effect 
that  wear  has  on  this  type  of  gearing  is  to  increase  the  backlash 
between  the  worm  and  the  worm-wheel  teeth.  It  was  also  found 
in  these  tests  that,  as  far  as  noise  was  concerned,  the  ball  bearings 
used  in  the  design  did  not  run  anywhere  near  as  quietly  as  the 
worm  and  worm-wheel,  which  would  indicate  that  from  this 
point  of  view  the  conditions  met  with  in  this  type  of  gearing  are 
almost  ideal.  This  type  of  worm-gearing  has  also  proved  highly 
satisfactory  for  reduction  gearing  in  connection  with  electric 
motors.  A  particularly  good  example  which  illustrates  the 
adaptability  of  this  type  of  gearing  to  large  reductions  was  a 


'METHODS   OF  CUTTING  TEETH  227 

reduction  gear  having  a  ratio  of  451.5  to  i  that  showed  an  effi- 
ciency of  80  per  cent  when  transmitting  10  horsepower  under  tests 
covering  a  considerable  period  of  time. 

Various  Methods  Compared.  —  Each  of  the  various  methods 
of  cutting  worm-wheel  teeth  described  has,  however,  its  field  of 
usefulness.  Gashing,  as  we  have  seen,  is  applicable  either  to 
cheap,  rough-and-ready  work  on  the  one  hand  or,,  on  the  other 
hand,  to  the  cutting  of  worm-wheels  which  are  not  required  to 
transmit  a  great  amount  of  power,  but  in  which  the  highest  degree 
of  accuracy  is  required.  The  process  of  hobbing  previously 
gashed  blanks  requires  the  least  degree  of  specialization  in  the 
machinery  used,  the  ordinary  milling  machine  having  all  the 
movements  and  adjustments  required.  This  process  is  perhaps 
the  one  followed  in  most  shops  in  making  worm  gearing  of  small 
size.  The  arrangement  (such  as  shown  in  Figs.  2  and  3)  in  which 
the  work  and  the  hob  are  positively  geared  together  so  that  pre- 
vious gashing  is  not  required,  is  quicker  than  the  last-mentioned 
method,  but  requires  special  machines  or  attachments.  The 
fly-tool  method  requires  a  still  more  elaborate  machine,  but  is 
the  least  expensive  of  all  in  the  matter  of  cutting  tools.  A  large 
hob  is  an  exceedingly  costly  appliance,  and  raises  the  cost  of 
production  to  an  alarming  degree,  particularly  when  but  one 
worm-wheel  has  to  be  cut.  The  use  of  a  simple  fly-cutter,  which 
may  be  ground  accurately  to  size  after  hardening  so  that  all 
inaccuracies  are  avoided,  is  thus  the  cheapest  as  well  as  the  most 
accurate  means  of  cutting  a  large  worm-wheel.  Where  many 
large  wheels  of  the  same  size  are  to  be  cut,  the  taper  hob  method 
is  the  most  satisfactory  one.  Hobbing  by  this  method  is,  of 
course,  more  rapid  than  by  the  fly- tool  process  employed  on  the 
same  machines,  though  the  latter  is  not  a  tedious  operation  by 
any  means,  as  a  solidly  supported  and  powerfully  driven  tool 
can  be  given  a  heavy  feed,  taking  off  chips  of  considerable  thick- 
ness. 

The  Worm.  —  The  methods  followed  in  cutting  the  other 
member  of  this  form  of  gearing,  the  worm,  have  already  been 
described  in  connection  with  the  methods  for  cutting  helical  and 
herringbone  gears.  A  few  general  remarks  may,  however,  be 


228 


WORM    GEARING 


-292... 


added.  Worm- threads  have  an  included  angle  between  the  sides 
of  29  degrees,  as  shown  by  the  sectional  view,  Fig.  9.  The  width 

of  the  worm-thread  tool  at 
the  end  equals  the  linear 
pitch  P  of  the  worm  (or  cir- 
cular pitch  of  the  gear)  mul- 
tiplied by  0.31.  It  is  difficult 
to  thread  worms  having  a 
large  lead  or  "quick  pitch" 

Fig.  9.    Dimensions  of  the  Worm  Thread      Qn  an  ordinary  Iatn6j  because 

the  lead-screw  must  be  geared  to  run  several  times  faster  than 
the  spindle,  thus  imposing  excessive  strains  on  the  gearing.  A 
common  method  of  overcoming  this  difficulty  is  to  mount  a  belt 
pulley  on  the  lead-screw,  beside  the  change  gear  and  belt  it  to 


TOOLS-REPRESENTING  TEETH  OF  WHEEL 


Fig.  10.     Method  of  Cutting  Hindley  Worm  with  Rotary  Cutter 
having  Teeth  Corresponding  with  Those  of  the  Wheel 

the  countershaft;  the  spindle  is  then  driven  through  the  change 
gearing. 

It  is  quite  common  practice  to  use  a  thread  milling  machine 
for  cutting  worms.     By  means  of  this  machine  worm  milling 


METHODS  OF   CUTTING   TEETH 


229 


becomes  quite  an  easy  matter.  The  machines  are  simple  in 
operation  and  good  work  can  be  turned  out  cheaply  and  satis- 
factorily. The  cutter  head  is  accurately  graduated  and  can  be 
swiveled  either  way  to  the  correct  angle  of  the  thread.  The 
headstock  spindle  is  hollow,  allowing  work  to  pass  completely 
through  it,  and  the  cross-slide  is  provided  with  an  automatic 
stop.  The  machines  can  also  be  stopped  automatically,  at  the 
end  of  a  thread,  thus  insuring  regularity  of  length  when  cutting 


Machinery 


Fig.  ii.     Cutting  the  Teeth  of  the  Hindley  Worm-wheel  with  a  Hob 
Corresponding  to  the  Worm 

multiple-pitch  worms.  There  is,  of  course,  as  is  evident  in  all 
milled  spiral  flutes,  a  tendency  to  leave  the  face  of  the  worm- 
thread  cut  somewhat  concave.  In  order,  therefore,  to  produce 
the  best  results,  one  maker  of  worm  drives  finds  it  advisable  to 
rough  out  the  worm  on  the  thread  milling  machine,  and  leave  a 
grinding  allowance  of  o.oio  inch  on  each  face  of  the  thread. 
The  worm,  after  hardening,  is  then  ground  by  a  special 
machine  which  corrects  this  concavity  of  the  worm  tooth  and 
also  removes  any  distortion  which  is  likely  to  be  caused  by 
hardening. 


230  WORM  GEARING 

Manufacture  of  the  Hindley  Worm  Gearing.  —  Any  positively 
operated  worm-wheel  bobbing  attachment  or  machine  may  be 
used  for  cutting  Hindley  worm  gearing.  The  manufacture  be- 
gins with  the  cutting  of  the  worm,  which  is  done  as  shown  in 
Fig.  10.  The  blank  is  mounted  on  the  spindle  of  the  machine 
ordinarily  occupied  by  the  hob,  while  a  large  diameter  disk  pro- 
vided with  cutting  tools  clamped  to  its  face  is  mounted  in  place 
to  represent  the  worm-wheel.  The  cutting  tools  mounted  on 
this  disk  each  represent  a  tooth  of  the  wheel,  being  of  the  same 
shape  and  cutting  on  the  same  diameter.  They  are  clamped  to 
the  face  of  the  disk  in  such  a  way  that  the  whole  arrangement 
represents  accurately  a  central  section  of  the  worm-wheel,  of 
which  (in  this  particular  case)  only  every  other  tooth  is  used. 
This  cutter  and  the  worm  to  be  cut  are  geared  together,  and 
slowly  fed  toward  each  other  as  when  hobbing  worm-wheels. 
The  teeth,  cutting  deeper  and  deeper  into  the  blank,  finally 
form  it  into  the  characteristic  "hour-glass"  shape  of  the  Hindley 
worm. 

In  cutting  the  wheel  the  process  is  reversed,  as  shown  in 
Fig.  ii.  A  hob  cut  in  the  same  way  as  the  worm  in  Fig.  10,  but 
with  its  teeth  relieved,  is  fed  into  the  wheel  blank  and  cuts  the 
teeth  in  a  way  exactly  identical  with  the  method  followed  in 
hobbing  worm-wheels,  the  only  difference  in  the  process  being 
the  difference  in  the  shape  of  the  hob  and  in  the  shape  of  the 
teeth  produced. 

The  above  paragraphs  relate  to  the  principles  involved  in  cut- 
ting Hindley  worm  gearing.  The  actual  operation  of  hobbing 
the  worm  and  the  gear  is  a  little  more  complicated  in  practice, 
as  certain  corrections  have  to  be  made  for  interference  that 
require  cutting  the  worm  first  with  a  cutter  head  of  the  same 
diameter  as  the  wheel  in  the  way  we  have  shown,  and  a  second 
time  with  a  head  of  somewhat  larger  diameter. 


CHAPTER  XI 
GASHING  AND  ROBBING  A  WORM-WHEEL 

IN  the  construction  of  worm  gearing  the  distance  from  the 
center  of  the  worm  to  the  center  of  the  worm-wheel  may  be 
fixed,  or,  in  some  cases,  variations,  within  reasonable  limits, 
may  be  permitted.  When  the  center  distance  is  fixed,  which 
will  be  the  condition  governing  the  work  under  consideration, 
the  mechanic  may  have  the  opportunity  of  testing  the  accuracy 
of  his  work  by  assembling  the  finished  gear  in  its  place,  which  is, 
of  course,  desirable.  We  shall  assume,  however,  that  in  this 
case,  such  opportunity  is  not  afforded. 

The  Gashing  and  Robbing  Process.  —  The  worm  itself 
should  first  be  accurately  finished  as  it  can  be  used  advanta- 
geously in  testing  the  center  distance  when  hobbing  the  worm- 
wheel.  We  shall  assume  that  this  has  been  done,  and  that  the 
wheel  blank  has  also  been  turned,  and  will  consider  the  method 
of  hobbing  the  teeth  in  the  latter  in  a  universal  milling  machine. 
It  is  first  necessary  to  gash  the  blank.  This  operation,  as  already 
indicated  in  the  preceding  chapter,  consists  of  cutting  teeth, 
which  are  approximately  the  shape  of  the  finished  teeth,  around 
the  periphery  of  the  blank,  by  the  use,  preferably,  of  an  involute 
gear  cutter  of  a  number  and  pitch  corresponding  to  the  number 
and  pitch  of  the  teeth  in  the  wheel.  If  a  gear  cutter  is  not  avail- 
able, a  plain  milling  cutter,  the  thickness  of  which  should  not 
exceed  three-tenths  of  the  circular  pitch,  may  be  used.  The 
corners  of  the  teeth  of  the  cutter  should  be  rounded,  as  other- 
wise the  fillets  of  the  finished  teeth  will  be  partly  removed. 
After  the  gashing  operation,  the  teeth  are  finished  to  conform  to 
the  shape  of  the  worm  by  revolving  the  blank  in  unison  with  a 
cutter  known  as  a  hob,  sinking  the  latter  into  the  blank  until  the 
teeth  are  cut  to  the  required  depth.  As  the  worm  which  meshes 
with  and  drives  the  worm-wheel  is  simply  a  short  screw,  it  will 

231 


WORM   GEARING 


be  apparent  that  if  the  axes  of  the  worm-wheel  and  worm  are  to 
be  at  right  angles  to  each  other,  the  teeth  of  the  wheel  must  be 
cut  at  an  angle  to  its  axis  in  order  to  mesh  with  the  threads  of 
the  worm.  The  method  of  setting  the  work  and  obtaining  this 
angle  will  first  be  considered. 

Setting  the  Work  and  the  Machine.  —  After  the  dividing  head 
and  tail-stock  have  been  clamped  to  the  table  and  the  cutter  has 


Machinery 


Fig.  i.     Setting  the  Milling  Machine  for  Gashing  a  Worm-wheel  Blank 

been  fastened  on  its  arbor,  the  table  is  adjusted  until  the  point 
of  the  center  of  the  dividing  head  and  the  center  of  the  cutter 
lie  in  the  same  vertical  plane.  This  may  be  done  by  moving  the 
carriage  in  or  out,  as  the  case  may  require,  until  the  center  of 
the  dividing  head  spindle  is  directly  under  the  center  of  the  cut- 
ter. If  a  standard  gear  cutter  is  used,  as  in  Fig.  i,  the  center  in 
the  head  may  be  set  to  coincide  with  a  center  line  on  the  cutter 
which  is  placed  there  by  the  makers  to  facilitate  setting  the 


METHODS  OF   CUTTING  TEETH 


233 


cutter  central  with  the  work  spindle.  If  a  plain  cutter  is  used 
(which  will  be  without  the  center  line),  a  convenient  method  of 
setting  it  is  to  place  an  arbor  on  the  head  and  tail  centers;  then 
with  the  blade  of  a  centering  square  projecting  upward,  adjust 
the  carriage  until  the  side  of  the  cutter  has  a  full  contact  with 
the  central  edge  of  the  blade.  A  second  adjustment  of  the  car- 
riage, equal  to  one-half  the  thickness  of  the  cutter,  will  locate 
the  latter  central  with  the  dividing  head.  When  the  table  is 
set  it  should  be  securely  clamped  to  the  knee  slide. 


x. 


Machinery 


Fig.  2.     Gashing  the  Worm-wheel  Blank 

The  blank  to  be  gashed  is  now  pressed  on  a  true-running  arbor 
which  is  mounted  between  the  centers  of  the  dividing  head  and 
tail-stock  as  illustrated  in  Fig.  2,  and  the  driving  dog  is  secured, 
to  prevent  any  vibration  of  the  work.  The  table  is  now  moved 
longitudinally  until  a  point  midway  between  the  sides  of  the 
blank  is  directly  beneath  the  center  of  the  cutter  arbor.  To 
set  the  blank,  place  a  square  blade  against  it  on  first  one  side  and 
then  the  other  and  adjust  the  table  until  the  distances  between 


234  WORM   GEARING 

the  blade  and  arbor,  on  each  side,  are  equal.  Of  course,  if  the 
diameter  of  the  arbor  were  greater  than  the  width  of  the  blank, 
the  measurements  would  be  taken  between  the  latter  and  the 
square  blade. 

Setting  the  Table  of  the  Machine.  —  The  table  should  now  be 
set  to  the  proper  angle  for  gashing  the  teeth.  This  angle,  which 
should  be  given  on  the  drawing,  may  be  determined  either  graph- 
ically or  by  calculation.  The  first  method  is  illustrated  in  Fig. 
3.  Some  smooth  surface  should  be  selected,  having  a  straight 
edge  as  at  A .  A  line  B,  equal  in  length  to  the  lead  of  the  worm 
thread,  is  drawn  at  right  angles  to  the  edge  A,  and  a  distance  C 
laid  off  equal  to  the  circumference  of  the  pitch  circle  of  the  worm. 


Machinery 


Fig.  3.     Method  of  Obtaining  Helix  Angle  of  Worm 

If  the  diameter  of  the  pitch  circle  is  not  given  on  the  drawing, 
it  may  be  found  by  subtracting  twice  the  addendum  of  the  teeth 
from  the  outside  diameter  of  the  worm.  The  addendum  equals 
the  linear  pitch  X  0.3183.  The  angle  a  is  then  accurately  meas- 
ured with  a  protractor,  as  shown  in  the  illustration. 

The  table  of  the  machine  is  then  swiveled  to  a  corresponding 
angle  which  can  be  measured  by  the  graduations  provided  on  all 
universal  milling  machines.  If  the  front  of  the  table  is  repre- 
sented by  the  edge  A,  and  the  worm  has  a  right-hand  thread, 
the  table  will  be  swiveled  as  indicated  by  the  line  ab\  if  the 
worm  has  a  left-hand  thread  the  table  will  be  turned  in  an  oppo- 
site direction.  The  angle  that  the  teeth  of  the  worm-wheel 
make  with  its  axis,  or  the  angle  to  which  the  table  is  to  be  swiveled, 
rriay  also  be  found  by  dividing  the  lead  of  the  worm  thread  by 
the  circumference  of  the  pitch  circle;  the  quotient  will  equal  the 


METHODS  OF  CUTTING  TEETH  235 

tangent  of  the  desired  angle.  This  angle  is  then  easily  found  by 
referring  to  a  table  of  natural  tangents. 

The  Gashing  Operation.  —  The  cutter  is  next  sunk  into  the 
blank  to  the  proper  depth.  If  the  diameter  of  the  cutter  is  no 
larger  than  the  diameter  of  the  hob  to  be  used,  the  depth  of  the 
gashes  should  be  just  a  trifle  less  than  the  whole  depth  of  the 
tooth.  This  whole  depth  is  found  by  multiplying  the  linear 
pitch  by  0.6866,  as  explained  in  Chapter  VI.  When  the  diam- 
eter of  the  cutter  is  greater  than  that  of  the  hob,  we  have  the 
condition  shown  at  F  in  Fig.  2.  The  depth  to  which  the  gashing 
cutter  should  be  set  is  then  limited  by  the  depth  of  the  cut  at  the 
side  of  the  blank.  Should  the  diameter  of  the  cutter  be  smaller 
than  that  of  the  hob,  the  condition  shown  at  G  is  encountered. 
Here  the  limiting  depth  to  which  the  cutter  should  be  set  is  shown 
to  be  on  the  center  line. 

Before  starting  a  cut,  bring  the  cutter  into  contact  with  the 
wheel  blank  and  set  the  dial  on  the  elevating  screw  at  zero. 
Then  sink  the  cutter  to  the  proper  depth,  as  indicated  by  the 
dial.  When  the  cutter  is  larger  than  the  hob,  the  depth  of  the 
tooth  should  be  laid  out  on  the  beveled  side  of  the  cutter  blank 
and  a  gash  cut  to  this  line.  The  depth  as  indicated  on  the  dial 
should  then  be  noted  and  all  the  gashes  cut  to  a  corresponding 
depth. 

The  Hobbing  Operation.  —  When  the  gashing  is  finished,  the 
table  is  set  at  right  angles  with  the  spindle  of  the  machine,  and 
the  cutter  is  replaced  with  a  hob  as  shown  in  Fig.  4.  The  out- 
side diameter  of  the  hob  and  the  diameter  at  the  bottom  of  the 
teeth  are  slightly  greater  than  the  corresponding  dimensions  of 
the  worm  in  order  that  there  may  be  clearance  between  the  latter 
and  the  worm-wheel.  Before  bobbing  the  dog  is  removed  from 
the  arbor. 

Adjust  the  table  longitudinally  until  the  center  of  the  blank 
is  directly  under  the  center  of  the  arbor  H.  The  blank  may  be 
set  quite  accurately  by  bringing  it  into  contact  with  the  arbor 
and  adjusting  the  work  until  the  arbor  rests  centrally  in  the 
throat  of  the  blank.  With  the  arbor  still  in  contact  with  the 
periphery  of  the  blank,  at  its  throat,  set  the  dial  of  the  elevating 


236 


WORM   GEARING 


screw  at  zero.  Measure  the  diameter  of  the  arbor  with  a  microm- 
eter, and  divide  this  dimension  by  2,  obtaining  the  radius. 
Measure  the  diameter  of  the  hob  K,  and  also  ascertain  its  radius. 
Subtract  the  radius  of  the  arbor  from  the  radius  of  the  hob, 
lower  the  knee  of  the  machine  an  amount  equal  to  this  result, 
and  again  set  the  dial  at  zero.  The  knee  may  now  be  lowered  a 
small  amount  in  order  that  the  blank  may  clear  the  hob  when  the 
latter  is  being  placed  on  the  arbor. 


/ 


Machinery 


Fig.  4.     Hobbing  the  Worm-wheel 

Tighten  the  hob  on  its  arbor,  and  then  raise  the  knee  until 
the  hob  is  in  mesh  with  the  gashes  in  the  blank.  It  will  be  ob- 
served that  the  whole  tooth  depth  has  not  been  reached  when 
the  hob  bottoms  in  the  gashes.  The  machine  is  now  set  in  mo- 
tion and  as  the  hob  revolves  the  blank  rotates  with  it.  The 
hob  is  now  fed  into  the  blank,  by  raising  the  knee,  until  the  dial 
indicates  the  correct  depth.  If  the  hob  is  properly  made,  and 
the  wheel  blank  accurately  sized,  the  teeth  will  be  cut  to  the 
correct  depth  when  the  inner  diameter  of  the  hob  grazes  the 
blank  at  its  throat  diameter.  The  hob  and  blank  should  now 
rotate  several  times  to  eliminate  any  spring  and  to  produce 
smooth  teeth. 


METHODS  OF   CUTTING  TEETH 


237 


After  the  work  has  made  a  few  revolutions,  to  insure  well- 
formed  teeth,  as  mentioned,  the  hob  and  wheel  are  disengaged, 
and  the  finished  worm  is  placed  in  mesh  with  the  latter,  as  shown 
in  Fig.  5,  after  the  chips  have  been  thoroughly  removed  from  the 
teeth  on  which  the  worm  bears.  The  worm  is  now  turned  along 
the  periphery  of  the  wheel  until  its  axis  is  parallel  with  the  top 
of  the  table.  It  may  be  set  in  this  position  by  testing  the  top 
surfaces  of  the  threads  at  either  end  with  a  surface  gage.  Set 
the  pointer  of  the  gage  so  that  it  just  touches  the  top  of  a  thread 


Fig.  5.  Determining  the  Center  Distance  Between  Worm  and  Wheel 

and  measure  the  distance  x  from  the  pointer  to  the  arbor.  Sub- 
tract from  this  dimension  the  difference  between  the  radii  r  and 
r\  of  the  arbor  and  worm,  and  the  result  will  be  the  center  dis- 
tance c.  If  the  worm  is  accurately  made  and  the  worm-wheel 
blank  turned  to  the  exact  dimensions,  this  center  distance  should 
be  very  close  to  the  distance  required.  If  necessary,  the  hob 
may  be  again  engaged  with  the  wheel  and  another  light  cut  taken. 
When  testing  the  center  distance,  as  explained  in  the  foregoing, 
it  is  better  to  lower  the  knee  sufficiently  to  make  room  for  the 
worm  beneath  the  hob,  and  not  disturb  the  longitudinal  setting 


238  WORM   GEARING 

of  the  table,  as  the  relation  between  the  wheel  and  hob  will  then 
be  maintained,  which  is  desirable  in  case  it  is  necessary  to  re- 
hob  the  wheel  to  reduce  the  center  distance. 

Reducing  Flats  on  Hobbed  Worm-wheel  Teeth.  —  The  larger 
the  diameter  of  a  hobbed  gear  —  the  pitch  remaining  the  same 
-  the  more  closely  the  tooth  outline  approaches  the  shape  of  a 
rack  tooth.  The  flats  left  on  the  teeth  by  the  hobbing  operation 
also  become  perceptibly  less  as  the  size  of  the  wheel  increases. 
The  flats  on  the  teeth  of  hobbed  worm-wheels  with  a  small  num- 
ber of  teeth  can  be  reduced  by  the  use  of  a  hob  having  a  large 
number  of  flutes.  Where  a  fly-cutter  hob  is  used,  the  flats  can 
be  further  reduced  by  moving  the  hob  along  its  axis  after  the 


Machineri 


Fig.  6.     Hob  Working  on  Worm-wheel  in  First  Position 

first  cut  has  been  taken  and  moving  the  worm-wheel  on  its  arbor 
a  corresponding  amount.  By  making  five  or  six  such  shifts  of 
the  hob,  a  very  smooth  worm-wheel  is  produced.  The  fly-cutter 
hob  and  gear  blank  must  be  geared  together  and  the  blank  cut 
to  depth  before  shifting  the  hob  on  its  axis  as  described.  . 

In  Fig.  6  the  tooth  D  of  gear  B  is  in  contact  with  the  hob  A 
at  the  point  F.  In  this  position,  a  series  of  flats  are  produced 
on  the  gear  as  it  revolves  in  the  direction  indicated  by  arrow  K. 
By  moving  the  hob  along  its  axis  a  distance  equal  to  a  small 
fraction  of  the  circular  pitch  in  the  direction  shown  by  the  arrow 
C,  the  hob  A  is  brought  into  contact  with  the  tooth  D  at  the  point 
H  as  indicated  in  Fig.  7.  A  new  series  of  flats  is  produced  in 
this  way  causing  the  corners  to  be  sheared  off  the  flats  which 


METHODS  OF   CUTTING  TEETH  2-39 

were  produced  when  the  hob  was  in  the  position  illustrated  in 
Fig.  6.  By  repeating  this  process,  moving  the  hob  along  its 
axis  a  number  of  times,  the  flats  produced  in  hobbing  a  worm- 
wheel  in  this  way  can  be  practically  eliminated.  The  total  dis- 
tance through  which  the  hob  is  moved  is  from  one  to  two  times 
the  circular  pitch.  The  movement  of  the  hob  along  its  axis 
can  be  accomplished  automatically  by  suitable  apparatus  prop- 
erly timed  and  operating  in  connection  with  the  driving  mechan- 
ism of  the  gear  and  hob.  Although  it  takes  slightly  longer  to 
hob  wheels  by  this  method,  the  increased  accuracy  of  the  work 
more  than  compensates  for  the  additional  time  which  is  necessary 


Machinery 


Fig.  7.     Conditions  after  Hob  has  been  Shifted,  showing  Change  in 
Relative  Position  of  Hob  and  Wheel 

for  the  operation.     This  method  of  gear  hobbing  is  used  by  the 
Boston  Gear  Works  when  a  smooth  worm-wheel  is  required. 

Suggested  Refinement  in  the  Hobbing  of  Worm-wheels.  — 
At  the  left  of  Fig.  8  is  a  sectional  view  showing  a  hob  in  the  act 
of  putting  the  last  finishing  touches  on  a  worm-wheel.  The  hob 
is  supposed  to  be  a  new  one  and  is  shown  in  the  condition  it  is  in 
when  first  received  from  the  makers.  At  the  right  of  Fig.  8  is 
shown  the  same  hob  putting  the  finishing  touches  on  a  worm- 
wheel  similar  to  that  in  the  first  case.  The  hob  in  this  case  is 
represented  as  having  been  in  use  for  a  considerable  time,  and 
having  been  ground  down  to  the  last  extremity,  ready  to  be 
discarded  for  a  new  one.  A  study  of  this  cut  will  show  that  if 
the  hob  is  made  in  the  first  place  to  properly  match  the  worm 
which  is  to  drive  the  wheel,  it  will  not,  when  worn,  cut  exactly 


240 


WORM   GEARING 


the  proper  form  of  tooth  in  the  blank  to  mesh  with  that  worm. 
The  teeth  are  cut  to  the  same  depth  in  each  case,  this  being 
necessary  in  order  to  make  a  proper  fit  with  the  worm,  which  is 
the  same  in  each  case  and  is  set  at  the  same  center  distance. 
The  grinding  away  of  the  worn  hob  has  reduced  its  diameter  by 
an  amount  indicated  by  dimension  b.  Its  center  is  therefore  at 
P  on  the  line  AB,  which  is  offset  by  a  distance  represented  by 
dimension  a  from  the  line  CD  on  which  the  center  0  of  the  new 
hob  is  located.  This  reduction  in  diameter  as  the  hob  is  ground 
away  from  time  to  time,  so  evidently  follows  from  the  construc- 
tion of  the  relieved  hob,  that  it  scarcely  needs  to  be  explained. 


Machinery 


Fig.  8.     The  Difference  in  Shape  of  Teeth  Cut  by  New  and  Old  Hobs 

It  is  said  of  relieved  hobs  that  they  can  be  ground  without 
changing  their  shape.  This  is  true  so  far  as  the  outline  of  the 
cutting  edge  is  concerned,  but  it  will  be  evident,  on  examining 
the  conditions  shown  at  the  right  hand  of  Fig.  8,  that  whatever 
the  outline  of  the  cutting  edges,  a  new  hob  of  radius  R  will  not 
cut  exactly  the  same  shape  teeth  in  the  blank  as  the  worn  hob 
with  radius  r.  The  elements  of  the  tooth  surface  it  generates 
are  struck  from  a  center  P,  removed  by  dimension  a  from  center 
O'  which  is  the  location  of  the  axis  of  the  worm  with  which  it 
meshes. 

It  is  possible,  and  perhaps  practicable,  to  overcome  this  slight 


METHODS  OF  CUTTING  TEETH 


241 


error;  that  is,  to  so  design  and  use  the  hob  that  it  will  cut  as 
correct  teeth  when  worn  as  when  new.  In  Fig.  9  dotted  line 
A  A  represents  the  outlines  of  a  new  hob  in  the  act  of  finishing 
the  worm-wheel  shown.  Were  a  hob,  ground  as  shown  at  the 
right  of  Fig.  8,  to  be  substituted  on  the  arbor  for  this  new  hob, 
without  altering  the  adjustment  of  the  machine  except  to  move 
the  hob  endwise  and  bring  it  in  contact  with  the  teeth  of  the 
wheel  on  one  side,  this  hob  would  be  represented  in  Fig.  9  by  the 
full  line  BB.  It  is  evident  that  the  left-hand  cutting  edges  of 
this  hob  coincide  (to  the  depth  they  extend  into  the  wheel) 
with  those  of  the  new  hob  represented  by  outline  A  A .  They  will, 
therefore,  so  far  as  they  extend,  cut  identically  similar  and  correct 
tooth  curves  with  the  new  hob. 


Fig.  9.  Cutting  Action  of  Ordinary 
Hob  at  Fixed  Center  Distance 
when  New  and  when  Worn 


Fig.   10.     Action   of   Proposed   Hob 
when  New  and  when  Old 
Graphically  Shown 


Teeth  cut  with  this  worn  hob  would,  however,  evidently  have 
two  faults.  The  space  would  be  too  narrow  at  the  pitch  line 
by  a  distance  measured  by  dimension  w,  and  they  would  not  be 
cut  deep  enough  in  the  blank  by  a  distance  measured  by  dimen- 
sion n.  The  problem  is  to  so  alter  the  design  and  application  of 
the  hob,  that,  even  when  worn,  we  can  cut  the  teeth  deep  enough 
and  the  space  wide  enough. 

Fig.  10  shows  these  conditions  fulfilled.  Dotted  line  CC  shows 
the  outline  of  the  proposed  hob  when  new.  The  only  difference 
between  the  proposed  hob  and  the  regular  one,  whose  outlines 
are  shown  by  the  dotted  line  A  A  in  Fig.  9,  is  that  the  teeth  have 
been  lengthened  by  an  amount  equal  to  dimension  o.  The  hob 
is  fed  in  as  was  the  case  with  the  new  hob  in  Fig.  9  until  the  dis- 
tance between  its  center  line  and  that  of  the  blank  is  the  same  as 


242  WORM   GEARING 

that  between  the  center  line  of  the  worm  and  the  wheel  in  the 
finished  machine.  The  increase  in  radius,  then,  by  an  amount 
o,  makes  the  hob  cut  a  clearance  deeper  than  is  necessary  by  that 
amount.  In  a  spur  gear  this  would  doubtless  be  a  bad  thing, 
since  it  would  make  the  tooth  slenderer  and  therefore  weaker. 
A  worm-gear,  however,  if  designed  to  be  sufficiently  durable 
for  continuous  use,  is  almost  certain  to  be  several  times  stronger 
than  necessary,  so  that  the  slight  weakening  involved  in  the 
change  is  not  of  great  importance.  When  the  hob  is  worn  to 
the  shape  shown  by  the  full  outline  DD,  the  hob  is  evidently  of 
the  same  diameter  as  the  new  one  in  Fig.  9,  represented  by  dotted 
outline  A  A.  The  tooth  space,  however,  as  before  explained,  will 
be  too  narrow  by  the  amount  m  in  Fig.  9  or  p  in  Fig.  10.  To 
widen  it  out  sufficiently,  it  is,  therefore,  necessary,  after  the  hob 
has  been  fed  in  to  the  proper  depth,  to  still  continue  the  cut- 
ting action,  feeding  the  hob  endwise,  however,  until  it  has  been 
displaced  to  the  position  indicated  by  outlines  D'D'.  The  re- 
sulting tooth  is  evidently  identical  with  that  given  by  the  new 
hob  A  A  in  Fig.  9. 

It  will  be  understood  that  when  the  hob  in  Fig.  10  is  new,  it 
will  not  have  to  be  shifted  end- wise  at  all,  since  it  will  cut  a  tooth 
space  of  the  proper  width  as  soon  as  fed  to  depth.  It  will,  how- 
ever, cut  a  space  deeper  than  necessary  by  an  amount  o.  The 
worn  hob,  on  the  other  hand,  has  to  be  shifted  longitudinally  by 
an  amount  p  and  cuts  to  exactly  the  required  depth.  These 
represent  the  two  extreme  conditions.  When  the  hob  is  half 
worn,  the  excess  clearance  will  be  equal  to  half  of  o,  and  the 
longitudinal  displacement  necessary  will  be  equal  to  half  of  p. 

While  the  change  in  the  design  of  the  hob  could  be  made 
easily  enough,  there  is  doubtless  some  difficulty  in  making  the 
required  change  in  the  nobbing  of  the  blank.  Taking  it  for 
granted  that  the  hob  has  been  made  to  suit  the  worm  which  is 
to  be  used,  and  that  it,  therefore,  has  the  same  pitch  diameter 
and  thickness  of  tooth  at  the  pitch  line,  the  method  of  procedure 
will  invariably  require  that  the  hob  be  fed  in  to  the  worm-wheel 
blank  until  the  distance  from  the  center  of  the  hob  to  that  of 
the  wheel  is  the  same  as  the  distance  from  the  center  of  the  worm 


METHODS  OF  CUTTING  TEETH  243 

to  that  of  the  wheel  in  the  finished  machine.  This  will  be  true 
whether  the  hob  is  new  or  worn,  and  whatever  may  be  the  kind 
of  machine  on  which  the  hobbing  is  done. 

The  method  by  which  the  hob  is  displaced  longitudinally  will 
depend  on  the  machine  used  for  the  operation.  There  will  be 
no  possible  way  of  doing  it  if  the  wheel  is  being  finished  while 
running  loosely  on  centers,  as  is  common  practice  when  the 
blank  has  first  been  gashed.  It  is  required  that  the  hob  and  blank 
be  positively  geared  together.  If  a  positively  driven  hobbing 
attachment  in  the  milling  machine  is  being  used,  the  matter  is 
simple.  If  the  hob  is  being  driven  by  the  spindle  of  the  machine, 
throw  in  the  cross  feed  in  either  direction  until  the  required 
longitudinal  displacement  of  the  wheel  with  relation  to  the  hob 
has  taken  place.  The  question  as  to  when  this  has  taken  place 
may  be  decided  either  by  measuring  the  thickness  of  the  tooth, 
as  in  cutting  spur  gears,  or  by  trying  the  wheel  from  time  to  time 
with  its  worm,  the  two  parts  being  mounted  in  place  in  the  ma- 
chine they  are  to  go  in,  or  held  the  proper  distance  apart  by 
other  means. 

For  regular  hobbing  machines,  as  at  present  made,  the  matter 
is  more  difficult.  The  required  longitudinal  displacement  of 
the  hob  may  be  obtained,  in  effect,  by  a  rotary  displacement  of 
the  hob  which  may  be  accomplished  by  slipping  (a  tooth  at  a 
time),  the  teeth  of  the  change  gears  connecting  the  hob  and  the 
blank.  If  a  hobbing  machine  were  to  be  built  especially  for  use 
in  the  way  which  is  here  suggested,  differential  gearing  could  be 
introduced  in  the  train  between  the  hob  and  the  wheel,  to  which 
a  power  feed  could  be  given  to  effect  the  rotary  displacement 
when  the  hob  has  been  fed  to  depth,  or  a  power  feed,  might  be 
applied  to  feed  the  spindle  and  its  attached  hob  endwise  to  effect 
the  same  result. 

It  is  not  certain  that  the  error  which  exists  in  the  use  of  re- 
lieved hobs  is  of  enough  importance  to  warrant  taking  any  trouble 
to  remedy  it.  It  is  always  well,  however,  to  know  and  under- 
stand such  errors  as  may  exist  in  any  process  of  this  sort,  no 
matter  if  they  are  of  no  great  practical  importance. 


CHAPTER  XII 
HOBS  FOR  WORM-GEARS 

IF  a  worm  and  gear  of  standard  proportions  are  brought  into 
mesh,  we  have  at  the  bottom  of  both  the  thread  of  the  worm 
and  teeth  of  the  gear  a  clearance  equal  to  one-tenth  of  the  thick- 
ness of  the  thread  or  tooth  at  the  pitch  line.  The  clearance  at 
the  root  of  the  gear  tooth  is  obtained  by  enlarging  the  hob  over 
the  diameter  of  the  worm,  by  an  amount  equal  to  two  clear- 
ances, while  the  clearance  of  the  tooth  in  the  thread  bottom  is 


Machinery 


Fig.  i.     Dimensions  of  the  Worm 

taken  care  of  by  the  proper  sizing  of  the  gear  blank. 

Dimensions  of  Hob.  —  While  it  may  be  customary  practice 
to  make  the  hob  an  exact  duplicate  of  the  worm  except  in  the 
one  item  of  outside  diameter,  a  hob  proportioned  as  suggested  in 
Fig.  3  is  recommended  as  one  that  will  give  much  more  satis- 
factory results,  and  be  found  to  be  well  worth  any  additional 
trouble  in  construction  required  beyond  that  for  the  style  ordi- 
narily used.  The  peculiar  feature  of  this  hob  is  that  it  is  an 
exact  opposite  of  the  worm  with  respect  to  the  proportions  of 
the  thread  shape;  the  depth  below  the  pitch  line  in  one  case 

244 


HOBS  FOR  WORM-GEARS 


245 


being  equal  to  the  height  above  the  pitch  line  in  the  other.  The 
object  of  this  is  to  have  a  hob  that  will  form  the  complete  out- 
line of  the  tooth  and  make  it  absolutely  certain  that  the  standard 
proportions  of  tooth  and  clearance  are  obtained.  Thus,  should 
the  diameter  of  the  blank  be  large,  the  hob  will  trim  off  the  top 
of  the  gear  teeth  to  the  proper  length,  when  the  proper  center 
distance  is  maintained. 

There  is  another  point  that  is  generally  overlooked,  and  that 
is  the  necessity  for  having  the  corners  of  the  thread  rounded  over, 
and  for  providing  a  liberal  fillet  at  the  root 
of  the  thread.  The  radii  of  the  rounded 
corner  and  the  fillet  may  be  as  large  as  the 
clearance  will  allow,  which  would  be  one- 
twentieth  of  the  circular  pitch  of  the  thread. 

The  effect  that  this  fillet  and  rounded 
thread  has  on  the  shape  of  the  tooth  is 
due  to  the  fact  that  it  increases  the  quality 
of  the  gear  and  the  strength  of  each  indi- 
vidual tooth.  The  rounded  corner  on  the 
thread  points  does  away  with  any  ten- 
dency to  scratch  the  surface  of  the  tooth 
in  the  cutting  action,  and  leaves  a  much 
larger  fillet  at  the  root,  greatly  increasing  the 
strength.  The  fillet  at  the  bottom  of  the  thread  rounds  off  the 
top  of  the  tooth  in  the  worm-gear,  removing  any  burrs,  and 
leaving  a  nicely  finished  product.  This  fillet  also  removes  the 
dangerous  tendency  of  the  hob  to  develop  cracks  in  the  harden- 
ing process  —  a  common  source  of  trouble  even  where  care  is 
taken.  Fig.  i  shows  the  proportions  of  the  worm  in  comparison 
with  the  hob  in  Fig.  3. 

Thread  Tool  for  Hob.  —  In  forming  the  hob  much  can  be 
gained  by  making  a  special  form  tool  of  correct  proportion  that 
will  leave  no  chance  for  error;  the  only  dimension  needing  care, 
then,  is  the  diameter.  Such  a  tool  is  shown  in  Fig.  2.  The 
figure  is  dimensioned  by  formulas,  so  that  a  tool  for  any  pitch 
can  be  easily  proportioned  from  it.  This  tool  may  be  made  by 
using  a  gear  caliper  without  resorting  to  the  protractor,  or  the 


Fig.  2.  Dimensions  of 
Tool  for  Threading 
the  Hob  • 


246 


WORM  GEARING 


protractor  may  be  used  in  laying  out  the  angle.  This  tool  may 
be  made  without  side  clearance,  providing  that  the  sides  incline 
in  the  same  direction  and  at  the  same  angle  that  the  thread  takes, 
but,  under  ordinary  circumstances,  where  only  one  hob  is  to  be 
made,  little  is  gained  by  having  no  side  clearance.  The  clearance 
may  be  made  from  5  to  10  degrees  from  the  angle  of  the  thread. 
Grinding  a  tool  like  this  of  course  changes  its  form,  so  it  must 
not  be  used  indefinitely  in  making  large  numbers  of  similar 
hobs. 

Number  of  Flutes  in  Hobs.  —  The  number  of  flutes  that 
should  be  provided  in  the  hob  is  a  point  on  which  very  little  is 
said,  various  authorities  differing  widely.  Where  the  hob  is  to 


v 


Machinery 


Fig.  3.     Dimensions  of  the  Worm-wheel 
Hob 


Fig.  4.    Dimensions  for  Flut- 
ing the  Hob 


be  used  in  an  automatic  hobbing  machine  in  which  the  hob  and 
blank  are  positively  geared  together,  the  number  of  flutes  may 
be  a  comparatively  small  number  as  compared  with  a  hob  that 
is  to  be  used  in  connection  with  ordinary  processes  of  hobbing 
worm  gears.  In  the  process  in  which  the  previously  gashed 
worm-gear  blank  is  swung  loosely  on  centers  and  revolved  by 
the  hob  as  the  latter  rotates,  the  hob  should  have  a  larger  num- 
ber of  flutes. 

A  rule  that  checks  up  well  with  present  practice  is  as  follows: 

To  find  the  number  of  flutes  in  a  hob,  multiply  the  diameter  of 
the  hob  by  three,  and  divide  by  twice  the  circular  pitch. 

The  above  rule  gives  suitable  results  on  hobs  for  general  pur- 
poses. 

Some  authorities  on  worm-gearing  state  that  the  number  of 


HOBS  FOR  WORM-GEARS  247 

flutes  in  a  hob  should  in  no  case  be  an  exact  multiple  of  the  num- 
ber of  threads.  Their  reason  for  this  rule  is  that  the  hob  so 
gashed  will  produce  a  much  smoother  tooth  and  one  nearer  correct 
in  shape,  because  no  tooth  in  the  hob  passes  the  same  tooth  in  the 
gear  twice  in  succession,  so  that  any  imperfections  in  the  shape 
of  the  individual  hob  teeth  are  counteracted  by  one  another. 
Another  authority  is  strong  in  his  advice  not  to  have  the  circum- 
ferential distance  from  flute  to  flute  equal  to  or  equally  divisable 
by  the  circular  pitch,  for  the  same  reason  as  stated  regarding 
the  former  rule.  From  these  statements  it  is  seen  that  to  obtain 
a  rule  that  would  be  at  once  simple  and  yet  take  all  conditions 
into  consideration,  would  be  a  difficult  proposition.  It  seems, 
however,  that  only  the  first  of  these  two  rules  is  a  logical  one. 
Owing  to  the  fact  that  hobs  have  teeth  only,  instead  of  full  sur- 
faces matching  the  worm,  the  curved  outlines  of  the  wheel  teeth 
are  merely  approximated  by  a  series  of  tangents.  If  the  num- 
ber of  flutes  in  the  hob  is  a  multiple  of  the  number  of  threads, 
the  hob  teeth  will  "track"  after  each  other,  giving  wheel  teeth 
only  roughly  approximated  by  a  comparatively  small  number  of 
long  tangents.  This  subject  is  treated  in  detail  in  the  latter 
part  of  this  chapter. 

•  Character  of  Flutes.  —  The  cutter  used  in  gashing  the  hob 
should  be  about  f  inch  thick  at  the  periphery  for  hobs  of  ordinary 
pitch,  while  for  those  of  coarser  pitch  a  cutter  J  inch  thick  would 
be  much  better.  The  width  of  the  gash  at  the  periphery  of  the 
hob  should  be  about  two-fifths  the  pitch  of  the  flutes.  The 
cutter  should  be  sunk  into  the  blank  so  that  it  reaches  from  ^3g- 
to  I  inch  below  the  root  of  the  thread.  Fig.  4  shows  an  end  view 
of  a  hob  gashed  according  to  these  rules. 

Where  a  hob  is  to  be  used  to  any  great  extent,  and  is  subject 
to  much  wear,  it  would  be  advisable  to  increase  the  diameter 
over  the  dimensions  given  from  o.oio  to  0.030  inch,  according 
to  its  diameter  and  pitch,  to  allow  for  decrease  in  diameter  due 
to  the  relief,  and  caused  by  grinding  back  the  cutting  face  in 
sharpening. 

Hobs  are  generally  fluted  parallel  with  the  axis,  but  it  is  obvi- 
ous that  they  should  be  gashed  on  a  spiral  at  right  angles  with 


248 


WORM   GEARING 


the  thread  helix  in  order  that  the  cutting  face  may  be  presented 
with  theoretical  correctness;  but  the  trouble  encountered  in 
relieving  the  teeth  on  the  ordinary  backing-off  attachment  is  the 
cause  of  the  common  mode  of  fluting.  When  the  pitch  or  lead 
is  coarse  in  comparison  with  the  pitch  diameter  of  the  hob,  so 
that  the  angle  is  correspondingly  steep,  it  may  be  best  to  flute 
on  the  normal  helix,  and  if  the  hob  cannot  be  machine  relieved, 
it  may  be  backed  off  by  hand. 

The  amount  of  relief  depends  much  on  the  use  for  which  the 
hob  is  intended.     A  hand  hob  for  hobbing  a  gear  in  position  may 


Machinery 


Fig.  5.     Diagram  for  the  Derivation  of  Formula  for  Spirally-fluted 

Hobs 

be  made  with  little  or  no  relief,  while  hobs  used  on  hobbing  ma- 
chines may  have  much  more  relief  than  those  used  on  the  milling 
machine. 

Spiral-fluted  Hob  Angles.  —  It  is,  of  course,  desirable  that 
hobs  should  be  fluted  at  right  angles  to  the  direction  of  the 
thread.  Sometimes,  however,  it  is  necessary  to  modify  this 
requirement  to  a  slight  degree,  because  the  hobs  cannot  be  re- 
lieved unless  the  number  of  teeth  in  one  revolution,  along  the 
thread  helix,  is  such  that  the  relieving  attachment  can  be  prop- 
erly geared  to  suit  it.  In  the  following  it  is  proposed  to  show 
how  an  angle  of  flute  can  be  selected  that  will  make  the  flute 
come  approximately  at  right  angles  to  the  thread,  and  at  the 
same  time  the  angle  is  so  selected  as  to  meet  the  requirements 
of  the  relieving  attachment. 


HOBS  FOR  WORM-GEARS  249 

Let  C  =  pitch  circumference; 

T  =  developed  length  of  thread  in  one  turn; 

N  =  number  of  teeth  in  one  turn  along  thread  helix; 

F  =  number  of  flutes; 

a  —  angle  of  thread  helix. 
Then  (see  Fig.  5)  : 

C  -T-  F  =  length  of  each  small  division  on  pitch  circumference. 
(C  -r-  F)  X  cos  a  =  length  of  division  on  developed  thread. 
C  +  cos  a  =  T. 

T  F 

Hence  ,— ^ =  N  = 


(C  -5-  F)  cos  a  cos2  a 

Now,  if  a  =30  degrees,  N  =  i^F', 
a  =  45  degrees,  N  =  2  F; 
a  =  60  degrees,  N  =  4  F. 

In  most  cases,  however,  such  simple  relations  are  not  obtained. 
Suppose  for  example  that  F  =  7,  and  a  =  35  degrees.  Then 
N  =  10.432,  and  no  gears  could  be  selected  that  would  relieve 
this  hob.  By  a  very  slight  change  in  the  spiral  angle  of  the 
flute,  however,  we  can  change  N  to  10  or  io|;  in  either  case  we 
can  find  suitable  gears  for  the  relieving  attachment. 

The  rule  for  finding  the  modified  spiral  lead  of  the  flute  is: 

Multiply  the  lead  of  the  hob  by  F,  and  divide  the  product  by  the 
difference  between  the  desired  value  of  N  and  F. 

Hence,  the  lead  of  flute  required  to  make  N  =  10  is: 

Lead  of  hob  X  (7  -*•  3). 

To  make  N  =  loj,  we  have: 

Lead  of  flute  =  lead  of  hob  X  (7  -J-  3.5). 

From  this  the  angle  of  the  flute  can  easily  be  found. 

That  the  rule  given  is  correct  will  be  understood  from  the  fol- 
lowing consideration.  Change  the  angle  of  the  flute  helix  |8 
so  that  AG  contains  the  required  number  of  parts  N  desired. 
Then  EG  contains  N  -  F  parts;  but  cot/3  =  BD  -=-  ED,  and 
by  the  law  of  similar  triangles : 

BD  =  j-X  BG,  and  ED  =  ^~ C. 
The  lead  of  the  spiral  of  the  flute,  however,  is  C  X  cot  0. 


250  WORM   GEARING 

Hence,  the  required  lead  of  spiral  of  the  flute: 


This  simple  formula  makes  it  possible  always  to  flute  hobs 
so  that  they  can  be  conveniently  relieved,  and  at  the  same  time 
have  the  flutes  at  approximately  right  angles  to  the  thread. 

Graphical  Method.  —  The  angle  of  the  flutes,  determined  so 
as  to  avoid  difficulties  in  relieving,  may  be  found  graphically  as 


B 
Machinery  N.Y. 


Fig.  6.  Graphical  Method  for  finding  Gashing  Angle  and  Number  of 
Flutes  for  which  Backing-off  Attachment  should  be  Set  for 
Spirally-fluted  Hobs 

follows:  First,  lay  off  a  base  line  AA7  Fig.  6,  of  any  convenient 
length.  -  Then  erect  the  perpendicular  AC  making  it  equal  to 
the  developed  length  of  the  pitch  circumference  of  the  hob. 
From  C  draw  line  CD  parallel  to  the  base  line  A  A  and  of  a  length 
equal  to  the  lead  of  the  hob.  Now  draw  diagonal  AD  which 
represents  the  thread.  Divide  AC  into  as  may  equal  parts  as 
there  are  flutes  in  the  hob,  as  a,  b  and  c.  From  C  and  a  draw 
lines  through  and  at  right  angles  to  the  diagonal  AD,  as  CE  and 
aF.  Then  length  EF  equals  the  pitch  of  the  flutes  on  the  thread 
when  the  gashing  is  at  right  angles  to  the  thread.  To  proceed, 
divide  AD  into  a  certain  number  of  equal  parts,  the  length  of 


HOBS  FOR  WORM-GEARS  251 

these  parts  to  be  as  near  to  the  length  EF  as  possible.  Step  off 
these  divisions  on  AD,  and  through  the  division  nearest  to  E, 
as  at  G,  draw  a  line  from  C  to  the  base  line  intersecting  the  base 
line  at  B.  This  line  CB  represents  the  gash,  line  AB  the  lead 
of  the  gash,  and  the  number  of  divisions  in  the  line  AD  equals 
the  number  of  flutes  to  one  revolution  of  the  hob,  for  which  we 
must  gear  the  machine. 

To  get  the  exact  length  of  AB,  divide  the  number  of  divisions 
in  A  G  by  the  number  of  divisions  in  GD  and  multiply  the  result 
by  the  length  of  the  line  CD  or  the  lead  of  the  hob.  The  angle 
a  which  is  the  angle  for  gashing  can  be  found  by  scaling  the 
diagram.  For  example,  let  the  hob  be  2  inches  pitch  diameter, 
lead  5  inches,  and  number  of  flutes  8. 

We  first  draw  base  line  AA,  and  the  line  AC  6.28  inches  long 
which  is  the  pitch  circumference.  Now  draw  CD  5  inches  long, 
and  then  draw  line  AD.  We  now  divide  AC  into  eight  equal 
parts  and  draw  lines  from  C  and  a  through  and  at  right  angles 
to  AD,  intersecting  AD  at  E  and  F.  Setting  the  dividers  to 
length  EF  we  step  off  line  AD  and  find  that  this  length  EF  will 
go  into  AD  SL  little  over  thirteen  times;  so  we  divide  this  line 
AD  into  thirteen  equal  parts.  It  is  now  necessary  to  gear  the 
machine  for  thirteen  flutes  to  one  revolution  of  the  hob. 

The  division  nearest  to  E  is  G,  so  by  drawing  a  line  from  C 
through  G  we  intersect  the  base  line  at  B.  In  the  line  GD  there 
are  five  divisions,  and  in  the  line  AG  there  are  eight  divisions. 
The  lead  of  the  hob  is  five  inches,  so  that  the  length  of  the  lead 
for  the  gash  or  AB  is  ^  X  $  =  8  inches.  By  measuring  on  the 
diagram  by  a  scale  we  find  the  gashing  angle  is  38^  degrees. 
Therefore,  we  will  gear  the  machine  used  in  backing-off  the  hob 
for  13  flutes  to  one  revolution,  and  we  will  gear  the  milling  ma- 
chine to  cut  a  lead  of  8  inches,  at  a  gashing  angle  of  38 J  degrees. 

Lengths  of  Worms  and  Hobs.  —  The  derivation  of  a  formula 
for  quickly  finding  the  approximate  length  of  a  hob,  or  of  a 
worm  to  run  with  a  worm-wheel  having  hobbed  teeth,  depends 
primarily  upon  an  assumed  relation  between  the  worm  and  worm- 
wheel  and  also  upon  the  pitch  and  size  of  the  wheel.  The  rela- 
tion between  the  length  of  worm  and  the  dimensions  of  the 


252  WORM  GEARING 

worm-wheel  differs  with  the  conditions  under  which  the  gears 
are  to  run  or  the  standards  of  the  manufacturer. 

In  Fig.  7  the  hob  (or  worm)  is  shown  extending  from  A  to  B, 
which  produces  a  hob  having  the  maximum  generating  action. 
This  length  also  provides  a  safe  allowance  for  any  end  adjust- 
ment which  may  be  necessary  for  the  worm.  A  hob  to  cut  a 
worm-wheel  without  interference  should  be  as  long  as  the  worm 
to  be  used,  and  neither  should  be  less  than  the  length  of  the  chord 
between  points  F  and  G,  which  are  on  the  wheel  pitch  circle. 
Let  AB  or  length  of  hob  =  /; 

BC  or  throat  circle  radius  =  r\ 
DE  or  whole  depth  of  tooth  =  d\ 
Number  of  teeth  in  worm-wheel  =  N. 
CE  =  BC  -  DE  =  r  -  d. 
Solving  the  right-angle  triangle  enclosed  by  the  lines  BC,  CE 


and  BE  for  r,  r  —  d  and-  J,  we  have: 


r2  -  2  rd  +  d2  +    -  =  r2. 
4 

-  =  2rd;  -  =  2rd- 

4  4 


In  order  to  further  simplify  the  formula,  we  will  assume  the 
pitch  to  be  i-inch  circular  pitch. 

DE  or  d  (whole  depth  of  tooth)  would  then  equal  0.6866  inch, 

and  BC  or  r  (throat  circle  radius)  would  be  equal  to 

2   X 

Substituting  we  get: 

r     /N  +  2       ~\ 

f  =  2  V  0.6866  ( -       |  -  0.6866 ) 
\3-i4i6  / 

This  can  be  simplified  as  follows: 


/  =  2  Vo.2i855  N  -  0.03432 


HOBS  FOR  WORM-GEARS 


253 


Squaring  both  sides  of  the  equation  we  get: 
/2  =  0.8742  N  —  0.13728. 

As  the  value  for/  need  be  only  approximate,  we  can  write  the 
equation  as  follows: 


8        8o' 


8 


8^  =  °; 


=o. 


8o/2-  joN 

Now,  if  we  solve  for  /  for  different  numbers  of  teeth,  the 
length  of  hob  or  worm  for  i-inch  circular  pitch  will  be  obtained. 


Machinery 


Fig.  7.     Diagram  of  Worm  and  Worm-wheel  for  Determining  Length 
of  Worm  and  Hob 

For  other  pitches  it  will  be  necessary  to  multiply  by  the  cir- 
cular pitch  to  obtain  the  correct  length. 

The  accompanying  table  gives  the  values  of  /  for  i  inch  cir- 
cular pitch.  To  illustrate  the  use  of  this  table,  suppose  we  de- 
sire to  find  the  length  of  a  worm  to  suit  a  f -inch  circular  pitch 
worm-gear,  having  39  teeth.  Find  the  value  for  /  in  the  table 
opposite  39  teeth.  This  value  is  5.83  and  multiplied  by  the 
pitch,  f  inch,  gives  4.37,  or  about  4!  inches,  which  is  the  length 
of  the  worm  or  hob. 


254  WORM   GEARING 

Table  of  Constants  for  Determining  the  Lengths  of  Worms  or  Hobs 


Factor/  equals  length  of  worm  when  circular  pitch  is  i  inch.     To  find 

length  for  any  other  pitch,  multiply  factor/  corresponding  to  given  num- 

ber of  teeth  in  worm-wheel  by  the  required  pitch. 

No.  of  Teeth  in 
Worm-wheel 

Factor  /for 
i-inch 
Circular  Pitch 

No.  of  Teeth  in 
Worm-wheel 

Factor  /for 
i-inch 
Circular  Pitch 

No.  of  Teeth  in 
Worm-wheel 

Factor  /for 
i-inch 
Circular  Pitch 

No.  of  Teeth  in 
Worm-wheel 

Factor  /for 
i-inch 
Circular  Pitch 

No.  of  Teeth  in 
Worm-wheel 

Factor  /for 
i-inch 
Circular  Pitch 

10 

2-93 

44 

6.18 

78 

8.25 

112 

9.92 

I46 

11.28 

II 

3-08 

45 

6.25 

79 

8.30 

H3 

9.96 

147 

11.32 

12 

3-22 

46 

6.32 

80 

8-35 

114 

IO.OO 

148 

11.36 

13 

3-35 

47 

6-39 

81 

8.40 

H5 

10.04 

149 

11.40 

14 

3-48 

48 

6.46 

82 

8-45 

116 

10.08 

150 

11.44 

15 

3-6o 

49 

6-53 

83 

8.50 

117 

10.12 

151 

11.48 

16 

3-72 

50 

6.60 

84 

8-55 

118 

10.16 

152 

11.52 

i? 

3-84 

5i 

6.67 

85 

8.60 

119 

IO.2O 

153 

11.56 

18 

3-95 

52 

6-74 

86 

8.65 

1  20 

10.  24 

154 

1  1.  60 

iQ 

4.06 

53 

6.80 

87 

8.70 

121 

10.28 

155 

11.64 

20 

4-17 

54 

6.86 

88 

8-75 

122 

10.32 

156 

11.68 

21 

4.27 

55 

6.92 

89 

8.80 

123 

10.36 

157 

11.72 

22 

4-37 

56 

6.98 

90 

8.85 

124 

10.40 

158 

n-755 

23 

4-47 

57 

7.04 

9i 

8.90 

125 

10.44 

159 

11.79 

24 

4-57 

58    ' 

7.10 

92 

8-95 

126 

10.48 

1  60 

11.825 

25 

4.66 

59 

7.16 

93 

9.00 

127 

10.52 

161 

11.86 

26 

4-75 

60 

7.22 

94 

9-°5 

128 

10.56 

162 

11.895 

27 

4-84 

61 

7.28 

95 

9.10 

129 

10.60 

163 

n-93 

28 

4-93 

62 

7-34 

96 

9-J5 

130 

10.64 

164 

11.965 

29 

5.02 

63 

7.40 

97 

9.20 

131 

10.68 

165 

12.  OO 

30 

5-ii 

64 

7.46 

98 

9-25 

132 

10.72 

166 

12.035 

31 

5.20 

65 

7-52 

99 

9-30 

133 

10.76 

167 

I2.O7 

32 

5-28 

66 

7.58 

100 

9-35 

134 

10.80 

168 

12.105 

33 

5.36 

67 

7.64 

IOI 

9.40 

135 

10.84 

169 

12.14 

34 

5-44 

68 

7.70 

IO2 

9-45 

136 

10.88 

170 

12.175 

35 

5-52 

69 

7.76 

I03 

9-50 

137 

10.92 

171 

12.21 

36 

5-6o 

70 

7.82 

IO4 

9-55 

138 

10.96 

172 

12.245 

37 

5-68 

7i 

7.88 

105 

9.60 

139 

II.  OO 

173 

12.28 

38 

5-76 

72 

7-94 

1  06 

9-65 

I4O 

11.04 

174 

12.315 

39 

5-83 

73 

8.00 

107 

9.70 

141 

11.08 

175 

12.35 

40 

5-90 

74 

8.05 

108 

9-75 

142 

II.  12 

176 

12.385 

4i 

5-97 

75 

8.10 

109 

9.80 

143 

ii.  16 

177 

12.42 

42 

6.04 

76 

8.15 

no 

9.84 

144 

11.20 

178 

12.455 

43 

6.  ii 

77 

8.20 

III 

9.88 

145 

11.24 

179 

12.49 

HOBS  FOR  WORM-GEARS  255 

Number  of  Flutes  in  Hobs.  —  The  question  of  how  many 
gashes  to  cut  in  a  worm  hob,  particularly  if  the  hob  is  multiple 
threaded,  has  always  been  a  puzzling  one  for  most  mechanics. 
Many  believe  that  the  only  requirement  is  that  the  number  of 
gashes  must  have  no  common  factor  with  the  number  of  threads 
in  the  worm.  That  is  to  say,  if  the  worm  is  quadruple  threaded, 
the  number  of  gashes  should  be  9  or  7  rather  than  8.  If  the  worm 
is  sextuple  threaded,  the  number  of  gashes  should  be  7  or  n 
rather  than  8,  9  or  10.  This  is  one  requirement,  but  there  seem 
to  be  other  factors  that  enter  into  the  decision  as  well.  These 
were  brought  to  the  attention  of  Mr.  R.  E.  Flanders,  who  care- 
fully investigated  this  subject  a  few  years  ago,  by  Mr.  N.  B. 
Chace,  superintendent  of  the  Cincinnati  Shaper  Co.,  who  was 
endeavoring  to  obtain  a  hob  that  would  cut  smooth,  regular 
teeth  for  the  worm-wheel  of  the  spindle  drive  in  a  machine  he 
was  building. 

The  worm-wheel  of  this  drive  had  35  teeth.  The  worm  had 
7  threads  and  a  lead  of  5  inches.  The  number  of  flutes  or  gashes 
in  the  hob  was  9.  These  gashes  were  milled  spirally  so  that  they 
were  at  right  angles  to  the  thread.  The  hob  was  made  by  a 
well-known  firm  which  makes  a  specialty  of  such  work;  it  was 
proved  by  subsequent  tests  to  be  accurately  and  finely  made, 
and  altogether  a  very  creditable  piece  of  work.  Do  what  he 
could,  however,  Mr.  Chace  was  unable  to  hob  worm-wheels 
that  would  be  satisfactory.  When  tried  in  place  in  the  machine 
and  run  with  the  worm,  each  one  appeared  to  have  five  low  spots, 
almost  as  if  the  pitch  line  were  a  pentagon  instead  of  a  circle. 
The  wheels  were  taken  out  and  the  thickness  of  the  teeth  in  the 
center  of  the  throat  at  the  pitch  line  measured  as  accurately  as 
possible. 

The  results  of  one  of  these  tests  are  shown  in  Fig.  8,  where  it 
will  be  seen  that  there  is  a  regular  recurrence  of  thin  teeth  in 
each  fifth  of  the  circumference  of  the  wheel  with  less  marked 
series  of  fine  intermediate  thin  teeth.  The  diagram  in  the 
center  of  the  figure  shows  graphically  (by  the  exaggerated  radial 
distance  from  the  center)  the  variation  in  the  measurements 
obtained.  The  first  thought  would  naturally  be  that  the  hob 


256 


WORM  GEARING 


had  warped  out  of  true  in  hardening,  in  which  case  the  ratio 
between  the  worm  and  the  wheel  of  5  to  i  would  give  the  error 
indicated;  but  careful  measurements  failed  to  detect  any  error 
of  this  kind,  either  in  the  periphery  of  the  hob  or  on  the  sides  of 
the  cutting  edges. 

The  Imperfect  Generating  Action  of  the  Hob.  —  To  find  what 
was  really  the  trouble  with  the  hob  (or  rather,  with  the  work  of 


Machinery 


Fig.  8. 


Thick  and  Thin  Teeth  Produced  by  Incomplete  Generating 
Action  on  the  Part  of  the  Hob 


the  hob,  for  the  hob  was  found  to  be  all  right),  it  will  be  neces- 
sary to  study  its  action  in  cutting  a  worm-wheel.  The  dia- 
gram in  Fig.  9  will  serve  to  illustrate  some  of  the  important 
points  connected  with  this  action.  In  the  upper  part  of  the  dia- 
gram at  the  right  is  shown  an  end  view  of  a  single-threaded  hob 
having  six  gashes.  To  the  left  of  this  is  shown  the  pitch  cylin- 
der of  the  hob  with  a  helix  traced  upon  it,  representing  the  center 


HOBS  FOR  WORM-GEARS 


257 


PITCH  CYLINDER  OF  HOB 
I  f 


Machinery 


Fig.  9.    Finding  the  Number  of  Cuts  per  Linear  Pitch 


258  WORM   GEARING 

of  the  thread.  Lines  parallel  with  the  axis  of  the  work  are 
drawn  on  this  pitch  cylinder,  representing  the  intersection  of  the 
faces  of  the  teeth  with  the  cylinder.  The  intersections  of  the 
helix  with  these  lines  at  #,  b,  c,  d,  etc.,  represent  the  positions  on 
the  pitch  cylinder  of  the  center  of  each  of  the  teeth  of  the  hob. 

Below  this  representation  of  the  pitch  cylinder  is  shown  a 
development  of  its  circumference  through  an  axial  length  equal 
to  the  linear  pitch  of  the  worm,  represented  in  this  case  by  the 
distance  ci.  On  this  development,  the  tooth  helix  between  c 
and  i  becomes  a  straight  line,  as  shown,  and  the  center  of  the 
tooth  faces  c,  J,  e,  /,  g,  h  and  i  are  developed,  as  before,  by  the 
intersection  of  this  tooth  line  with  equally  spaced  horizontal 
lines  representing  the  six  gashes  in  the  circumference.  Below 
this  development  of  the  circumference  of  the  hob  is  shown  a 
series  of  outlines  of  the  cutting  edges  of  the  hob,  each  one  of 
which  has  its  center  directly  below  the  corresponding  center 
Cidij  etc.,  in  the  development.  These  outlines  evidently  repre- 
sent the  successive  positions  of  the  teeth  of  the  hob  as  they  pass 
the  plane  of  the  throat  of  the  worm-wheel  in  hobbing  its  teeth. 
There  are  seven  of  these  positions,  but  as  one  of  them  belongs 
to  the  next  section  of  the  hob,  from  i  to  0,  the  diagram  shows 
six  positions  of  the  hob  teeth  in  the  linear  pitch  of  the  hob. 

This  means,  of  course,  that  the  hob  does  not  accurately  gen- 
erate a  tooth  of  the  wheel,  since  it  acts  on  it  only  in  the  six  suc- 
cessive positions  shown,  instead  of  continuously  throughout  the 
whole  distance  of  the  circular  pitch.  In  order  to  get  smooth 
accurate  teeth,  the  number  of  cuts  in  the  linear  pitch  must  be 
made  as  many  as  possible;  the  more  there  are,  the  more  nearly 
perfect  would  the  generating  action  be;  the  less  there  are,  the 
rougher  will  be  the  tooth.  Now,  as  will  be  shown  later,  there  is 
but  one  cut  per  linear  pitch  in  the  example  mentioned  in  the 
paragraphs  headed  "  Number  of  Flutes  in  Hobs."  Under  these 
circumstances  the  teeth  of  the  worm,  instead  of  being  smoothly 
generated  to  a  curve,  are  only  slabbed  out  by  a  series  of  flat  cuts, 
as  indicated  in  Fig.  10. 

The  reason  for  the  five  thin  teeth  in  the  circumference  is  now 
evident.  At  every  fifth  of  a  revolution  those  teeth  of  the  hob 


HOBS  FOR  WORM-GEARS 


259 


which  formed  the  outline  near  the  pitch  line  oi  the  gear  gave  it 
a  shape  similar  to  that  shown  at  the  right  of  the  engraving.  In 
the  thick  teeth  the  conditions  shown  at  the  left  are  found,  where 
the  outline  of  the  tooth  at  the  pitch  line  is  formed  by  cuts  so 
placed  as  to  make  corners  at  this  point  instead  of  flats  as  at  the 
right.  This  means  that  the  teeth  measured  on  the  pitch  line 
are  thick  at  one  point  and  thin  at  the  other,  giving  high  spots, 
as  found  by  running  the  wheel  with  the  worm,  and  as  indicated 
also  by  the  measurement  shown  in  Fig.  8.  The  reason  for  the 
intermediate  thin  spots  between  the  even  fifths  is  not  clear  from 


Machinery 


Fig.  10. 


Example  of  Thick  and  Thin  Teeth  due  to  Incomplete 
Generating  Action 


the  preceding  explanation,  but  they  are  doubtless  due  to  the 
particular  arrangement  of  the  flats  on  the  tooth  outline  which 
happens  in  this  particular  wheel. 

Diagrams  for  Finding  the  Number  of  Cuts  per  Linear  Pitch.  — 
It  is  evidently  a  simple  matter  to  draw  diagrams  for  any  case 
showing  the  development  of  one  linear  pitch  on  the  pitch  surface 
of  the  hob,  as  in  Fig.  9,  and  find  out  from  that  diagram  how  many 
cuts  the  hob  gives  in  that  distance.  In  Fig.  n  eight  such  dia- 
grams are  shown,  for  eight  different  cases.  The  first  case  is  a 
single-threaded  hob  having  five  gashes.  This  diagram,  which  is 
similar  to  the  one  in  Fig.  9,  shows  that  there  are  five  cuts  to  the 
linear  pitch.  In  the  second  diagram  a  hob  of  the  same  diameter 
and  the  same  linear  pitch  having  also  five  gashes,  but  quintuple 


260 


WORM  GEARING 


instead  of  single  threaded,  gives  but  one  cut  to  the  linear  pitch. 
This  is  evidently  a  very  bad  condition  and  one  to  be  avoided,  if 
possible,  and  it  is  evidently  brought  about  from  the  fact  that 
the  number  of  gashes  is  the  same  as  the  number  of  threads. 


CASE  II         CASE  in          CASE  IV  CASE  V         CASE  VI         CASE  VII        CASE  VIII 


5  CUTS      I        1CUT      I      6  CUTS     I    6  CUTS      I    54- CUTS   I    1  +  CUT      I   6  •»•  CUTS    I    7+  CU' 
1  THREAD  I  5  THREADS      1   THREAD  |5  THREADS      1   THREAD J5 THREADS      1  THREAD  \  5  THREA 


MacMn 

Fig.  ii.    Diagram  for  Finding  the  Number  of  Cuts  per  Linear  Pitch 

In  the  third  and  fourth  cases  the  number  of  gashes  has  been 
increased  to  six,  with  a  single  thread  in  one  case  and  a  quintuple 
thread  in  the  other.  In  each  case  there  are  six  cuts  to  the  linear 
pitch.  The  fifth  and  sixth  cases  are  the  same  as  the  first  and 
second,  except  that  the  lines  representing  the  gashes  have  been 


HOBS  FOR  WORM-GEARS  261 

drawn  at  right  angles  to  the  lines  representing  the  tooth  helices 
as  would  be  necessary  for  hobs  which  are  gashed  helically  in  a 
direction  normal  to  the  tooth  helices.  These  cases  will  be  seen 
to  correspond  to  Nos.  i  and  2  except  that  the  number  of  cuts  has 
been  increased  in  proportion  to  the  cosine  of  the  gashing  angle, 
so  that  we  have  5  +  cuts  for  Case  V,  and  i  +  cuts  for  Case  VI. 
In  Cases  VII  and  VIII  are  shown  the  same  conditions  as  in 
Cases  III  and  IV,  except  that  the  hob  is  gashed  helically.  In 
this  case,  also,  the  number  of  cuts  is  increased  in  inverse  propor- 
tion to  the  cosine  of  the  gashing  angle,  giving  6  +  and  7  +  cuts, 
respectively,  for  the  two  cases,  the  hobs  having  six  gashes  each. 

In  Fig.  12  are  shown  four  more  cases,  considerably  more  com- 
plicated than  those  in  Fig.  n.  Here  are  four  hobs,  all  of  the 
same  linear  pitch  and  pitch  diameter,  and  all  octuple  threaded, 
with  threads  of  the  same  lead  and  helix  angle,  the  only  differ- 
ence in  the  four  being  in  the  number  of  gashes  and  the  method 
of  cutting  them.  In  Cases  IX  and  XI  there  are  eleven  gashes, 
and  in  Cases  X  and  XII  there  are  twelve.  Cases  IX  and  X  are 
gashed  parallel  with  the  axis.  This,  of  course,  would  be  utterly 
impracticable  in  any  hob  having  threading  angles  as  great  as 
those  shown  here,  so  the  example  is  not  a  practical  one,  being 
used  only  for  the  sake  of  illustrating  a  principle.  Cases  XI  and 
XII  which  are  gashed  helically  and  normally  at  right  angles 
to  the  threads,  represent  what  would  be  the  practical  construc- 
tion of  these  hobs.  Projecting  the  intersections  of  the  thread 
lines  with  the  gash  lines,  down  to  the  bottom  of  each  diagram, 
we  get  for  Case  IX,  eleven  cuts  to  a  linear  pitch;  for  Case  X, 
three  cuts  to  a  linear  pitch;  for  Case  XI,  38  +  cuts;  and  for  Case 
XII  ii  +  cuts. 

The  Effect  of  the  Number  of  Teeth  in  the  Wheel.  —  There  is 
still  another  factor  entering  into  this  problem  —  the  number  of 
teeth  in  the  wheel.  This  is  the  factor  which  gave  so  much 
trouble  in  the  case  mentioned.  Take,  for  instance,  Case  IV  in 
Fig.  ii.  Suppose  that  the  quintuple-threaded,  six-gashed  hob, 
represented  by  that  diagram,  were  cutting  a  25-tooth  wheel,  it 
would  not  give  the  six  cuts  indicated  by  the  diagram.  The 
reason  for  this  will  appear  by  comparing  Case  IV  with  Case  III. 


262 


WORM   GEARING 


In  Case  III,  where  the  hob  is  single  threaded,  all  of  the  cuts  rep- 
resented by  the  points  of  the  intersections  of  the  thread  and 
gash  lines,  are  along  the  same  thread. 

In  Case  IV,  however,  each  of  the  five  thread  lines  in  the  dia- 
gram has  but  one  intersection.     That  means  that  if  the  number 


STRAIGHT  GASHES.  PARALLEL  WITH  AXIS  HELICAL  GASHES,  NORMAL  TO  TOOTH  HELICES 


Jfachineru,N>T» 
Fig.  12.    Diagram  for  Finding  the  Number  of  Cuts  per  Linear  Pitch 

of  teeth  in  the  gear,  as  in  the  supposed  example,  is  a  multiple  of 
the  number  of  threads  in  the  worm  or  hob,  each  of  those  threads 
will  come  back  into  the  same  tooth  spaces  in  the  wheel  at  each 
revolution  of  the  latter,  so  that  for  each  tooth  space  there  is  but 


HOBS  FOR  WORM-GEARS  263 

one  cutting  position  of  the  hob  tooth  —  that  represented,  for 
instance,  by  point  a  for  one  of  the  tooth  spaces,  point  b  for  the 
next,  c  for  the  next,  and  so  on.  If,  on  the  other  hand,  there  were 
26  teeth  in  the  wheel,  the  first  time  it  went  around,  point  a  would 
cut  in  a  certain  tooth  space;  the  second  time  around  point  b 
would  come  in  the  same  space,  and  the  third  time  around  point 
c  would  follow,  so  that  each  tooth  space  would  get  the  benefit 
of  each  one  of  the  six  cuts,  the  same  as  in  the  single- thread  worm 
for  Case  III.  It  is  thus  seen  that,  besides  the  other  points 
mentioned,  the  number  of  teeth  in  the  wheel  has  an  effect  on  the 
number  of  cuts  of  the  worm  per  linear  pitch.  In  the  practical 
case  previously  mentioned  there  was  a  35-tooth  worm-wheel  and 
a  y-threaded  worm,  giving  the  worst  conditions  possible. 

A  General  Formula  for  Determining  the  Number  of  Cuts.  — 
From  the  preceding  description  it  will  be  seen  that  there  are 
three  points  to  be  taken  into  consideration  in  determining  the 
number  of  cuts  per  linear  pitch  (and  the  consequent  generating 
efficiency  of  the  worm)  from  the  number  of  gashes  in  the  hob. 
These  factors  are:  First,  the  relation  of  the  number  of  threads 
of  the  hob  to  the  number  of  gashes.  Second,  the  angle  of  the 
gashing.  Third,  the  relation  of  the  number  of  threads  of  the  hob 
to  the  number  of  teeth  in  the  wheel  to  be  cut.  It  might  be  con- 
sidered that  there  is  a  fourth  factor,  that  of  the  absolute  number 
of  teeth  in  the  wheel,  since  the  trouble  that  comes  from  a  small 
number  of  cuts  per  linear  pitch  is  exaggerated  in  the  case  of  a 
wheel  having  very  few  teeth.  This  is  not  a  matter  of  calculation, 
however,  and  would  not  enter  into  the  calculations  anyway,  since 
for  any  given  case  for  which  a  hob  is  being  designed,  the  number 
of  teeth  in  the  wheel  is  determined  approximately  at  least. 

Now,  instead  of  drawing  diagrams  such  as  shown  in  Figs.  9, 
ii  and  12,  it  would  be  better  if  a  simple  mathematical  expression 
could  be  obtained  which  would  give  the  number  of  cuts  per 
linear  pitch  directly.  This  can  easily  be  done.  The  effect  of 
the  number  of  gashes  with  relation  to  the  number  of  threads  is 
as  follows:  The  number  of  cuts  per  inch  varies  inversely  with  the 
greatest  common  divisor  of  the  number  of  threads  and  the  number 
of  gashes  in  the  hob.  The  influence  of  the  number  of  teeth  in  the 


264  WORM   GEARING 

wheel  is  a  similar  one  and  may  be  expressed  as  follows:  The 
number  of  cuts  per  linear  pitch  varies  inversely  with  the  greatest 
common  divisor  of  the  number  of  threads  in  the  worm  and  the  num- 
ber of  teeth  in  the  wheel.  The  effect  of  the  angle  of  the  gashing 
may  be  expressed  as  follows  :  The  number  of  cuts  per  linear  pitch 
varies  inversely  with  the  square  of  the  cosine  of  the  gashing  angle, 
measured  from  a  line  parallel  with  the  axis  of  the  hob.  These 
statements  are  combined  in  the  following  formula: 

v  G 


in  which 

G  =  number  of  gashes; 
/3  =  angle  of  gashing  with  axis; 
D  =  G.  C.  D.  of  number  of  threads  and  number  of  gashes  in 

hob; 
D'  =  G.  C.  D.  of  number  of  threads  in  hob  and  number  of 

teeth  in  wheel; 

X  =  number  of  cuts  per  linear  pitch. 
(G.  C.  D.  =  "greatest  common  divisor.") 

It  is  easier  to  see  the  relationships  expressed  above,  from  the 
foregoing  diagrams  and  description,  than  it  is  to  explain  them. 
These  relationships,  although  quite  simple,  are  rather  elusive. 
Perhaps,  however,  the  effect  of  the  angle  will  be  understood 
from  the  figuring  of  the  triangle  at  the  base  of  the  diagram  for 
Case  XII.  Note  that  the  formula  is  true  only  for  the  usual  cases 
in  which  the  gashing  is  either  helical  and  normal  to  the  threading, 
or  straight  and  parallel  to  the  axis.  In  the  latter  case,  cos2  /3  =  i, 
since  /3  =  o  deg.,  and  the  effect  of  the  angle  disappears. 

Applying  the  formula  to  the  practical  example  already  men- 
tioned, we  have  the  following  values: 

G  =  g; 

)3  =  20  deg.  (assumed,  as  the  angle  was  not  given)  ; 
Z)  =  i  =  G.  C.D.  of  9  (number  of   gashes)  and  7  (number 

of  threads)  ; 

j)'  =  7  =  G.  C.  D.  of  7  (number  of  threads)  and  35  (number 
of  teeth  in  wheel). 


HOBS  FOR  WORM-GEARS  265 

Solving  for  the  number  of  cuts  per  inch,  we  have: 

X  =  9 

i  X  7  X  O.Q3972 

If  the  number  of  teeth  in  the  wheel  had  been  36  instead  of 
35,  the  number  of  cuts  would  have  been: 

v  9  9 

*  i  X  i  X  Q.93972  *  0.883  ~ 

which,  it  will  be  seen,  would  immeasurably  improve  conditions, 
giving  a  fine,  smooth  outline  for  this  number  of  teeth  in  the 
wheel.  In  the  actual  wheel,  as  cut  by  the  hob,  the  slab-sided 
effect  shown  in  Fig.  10  was  very  noticeable,  there  being  about 
three  cuts  to  each  face  of  the  tooth. 

Hobbing  Methods  which  give  a  Complete  Generating  Action. 
—  It  should  be  noted  that  while  this  faulty  generating  is  liable 
to  occur  with  hobbing  by  the  usual  method  of  sinking  the  cutter 
in  to  depth  in  a  blank,  the  same  difficulty  does  not  occur  in  the 
fly-tool  process  or  in  a  machine  using  a  taper  hob  fed  axially 
past  the  work,  as  described  in  a  preceding  chapter.  In  the  case 
of  these  machines,  working  with  either  taper  hobs  or  fly-cutters, 
the  number  of  cuts  per  pitch  is,  at  the  least  calculation,  the  num- 
ber of  revolutions  per  linear  pitch  of  advance  of  the  cutter 
spindle;  it  thus  runs  up  into  the  thousands,  where  the  diagrams 
shown  in  Figs.  9,  n  and  12  give  only  from  i  to  38. 

This  treatment  of  the  hob  question  takes  care  of  all  the  factors 
which  enter  into  a  determination  of  the  number  of  gashes  to  use 
in  a  hob,  so  far  as  this  effects  the  accuracy  of  the  generating 
action.  Expressed  briefly,  the  conclusions  are: 

Avoid  having  a  common  factor  between  the  number  of  threads 
and  the  number  of  gashes  in  the  hob. 

Avoid  having  a  common  factor  between  the  number  of  threads 
in  the  hob,  and  the  number  of  teeth  in  the  wheel. 


INDEX 


PAGE 

Abrasion  in  worm  gearing 183 

Addendum,  of  herringbone  gears A 85 

of  spiral  gears 7 5,  20 

of  worm  teeth 158, 166 

Angle,  center,  of  spiral  gears i 

helix,  of  worms 158, 166 

pressure,  of  herringbone  gears 82, 85 

spiral,  for  herringbone  gears 81, 85 

spiral,  for  worm  hobs 248,  250 

tooth,  of  spiral  gears 2 

worms  with  large  helix 167 

Automobile  drives,  worm  and  helical  gears  for 180 

Bearing  friction  in  worm  gearing 196 

Center  angle  of  spiral  gears i 

Center  distance,  of  herringbone  gears 84 

of  spiral  gears 4,  20 

of  worm  and  worm-gear 160, 166 

Change  gears  for  bobbing  machines,  calculating,  for  spiral  gears 123 

for  worm  gears 216 

Cost  of  herringbone  gears 70 

Cutters,  milling,  feed  marks  produced  by 137, 153 

Cutters  for  milling  spiral  gears 5,  20, 105, 107 

formula  for 25 

Depth  of  teeth,  in  herringbone  gears 85 

in  spiral  gears S,  20 

in  worms 158, 166 

Diameter,  herringbone  gears 85 

hobs  for  spur  and  spiral  gears 153 

hobs  for  worm-gears 244 

outside,  of  spiral  gears 6,  20 

pitch,  of  herringbone  gears 84, 85 

pitch,  of  spiral  gears 4,  20 

worm,  pitch  and  outside 158, 166 

worm-gears,  pitch  and  throat 158, 166 

worm-gears,  table  for  calculating 167, 168 

Differential  mechanism  of  gear-hobbing  machines,  advantages  of 134 

Drawing,  model  worm-gear 169 

267 


2  68  END-MILLS  —  GEARS 

PAGE 

End-mills  for  milling  herringbone  gears 76 

Efficiency  of  worm  gearing 170,  226 

tests 181 

theoretical 179 

Elevator  gearing 177, 183, 184 

Face  width  of  herringbone  gears f 81 

Feed  marks  produced  by  milling  cutters 137, 153 

Flats,  on  hobbed  worm-gear  teeth,  reducing. .' 238 

produced  by  gear  bobbing 144 

Flutes  in  worm-gear  hobs 246,  255 

general  formula  for  number  of 263 

Fly-tool  method  for  cutting  worm-gears 219 

Friction,  bearing,  in  worm  gearing 196 

Gashing  worm-gears 213,  231,  235 

Gashing  worm-gear  hobs 246,  255 

Gears,  action  of  herringbone. . .  •. 72 

action  of  spur 71 

advantages  of  herringbone 74 

applications  of  worm 161 

applied  to  automobile  drives,  worm  and  helical 180 

bearing  friction  of  worm 196 

change,  for  bobbing  spiral  gears 123 

change,  for  hobbing  worm  gears 216 

cost  of  herringbone 70 

cutters  for  spiral 5,  20, 105, 107 

cutting  of  Hindley  worm 230 

defects  in  hobbed ' 147 

efficiency  of  worm 170,  226 

efficiency  tests  on  worm 181 

examples  of  calculating  spiral 6, 19 

examples  of  calculating  worm 162 

fly-tool  method  for  cutting  worm 219 

for  freight  elevator,  worm 177, 183, 184 

formulas  for  dimensions  of  herringbone 85 

formulas  for  spiral 20,  29, 33 

formulas  for  worm 166 

gashing  and  hobbing  worm 213,  231,  235 

graphical  solution  of  spiral 8 

herringbone 69 

Hindley  worm 202,  230 

hobbing  process  applied  to  herringbone 78 

hobbing  spiral 118, 122, 137, 143 

hobbing  worm 216,  231,  235 

hobs  for  spiral 137 

hobs  for  worm 244 

horsepower  transmitted  by  herringbone 87, 92 


GEARS  —  HERRINGBONE  269 

PAGE 

Gears,  horsepower  transmitted  by  worm 187, 188 

load  and  efficiency  of  worm 170 

lubricants  for  worm .- 187 

material  for 89, 176 

safe  velocity  of  herringbone 89 

self-locking  worm 176, 191 

spiral : I 

strength  of  herringbone 86 

theoretical  efficiency  of  worm r 179 

worm 156 

Wuest  herringbone 69, 80 

Gear-cutting  machine,  universal in 

output  of 142 

Gear-hobbing  machines,  advantages  of  differential  mechanism 134 

calculating  gears  for 123,  216 

output  of 142 

Gear  teeth,  bobbing 118, 122,  216,  235 

hobbing,  flats  produced  by 144 

milling  spiral 101, 104,  no 

planing  or  shaping  spiral 98, 116 

reducing  flats  on  worm 238 

safe  load  on  worm 174 

Grant's  formula,  demonstration  of '. .       25 

Graphical  method  for  determining  spiral-fluted  hob  angles 250 

Graphical  solution  of  spiral-gear  problems 8 

Helical  gears i 

Helix  angle,  of  herringbone  gears 81, 85 

of  worm 158, 166 

worms  with  large 167 

Herringbone  gears 69 

action  of 72 

advantage  of 74 

application  to  steam  turbines 95 

cost  of 70 

face  width  of 81 

for  hydraulic  turbines 97 

for  machine  tools 96 

for  rolling  mills 97 

formulas  for  dimensions 85 

general  points  in  design 91 

hobbing  process  applied  to 78 

horsepower  transmitted 87, 92 

materials  for 89 

methods  of  forming  the  teeth  in 76, 98 

milling 76 

pitch  diameters  and  center  distances 84, 85 


270  HERRINGBONE  —  LINEAR 

PAGE 

Herringbone  gears,  planing 77 

pressure  angle 82, 85 

production  of 75 

safe  velocity *. 89 

spiral  angle 81, 85 

strength  of 86 

tooth  proportions 82, 85 

Wuest 69, 80 

Hindley  worm-gear 202 

cutting 230 

objections  to 209 

Hobs,  angles  of  spiral-fluted  worm-gear 248,  250 

diameters  of 153 

dimensions  for  worm-gear 244 

distortion  of 147 

flutes  in  worm-gear 246,  255 

for  spur  and  spiral  gears 137 

for  worm-gears 244 

general  formula  for  flutes  in  worm-gear 263 

length  of 251 

number  of  teeth  in 146,  246,  255 

shape  of  teeth  in 151 

taper,  for  cutting  worm-gear 223 

thread  tool  for 245 

Hobbed  gears,  defects  in 147 

reducing  flats  on  teeth 238 

Robbing 78, 118, 137,  216,  231 

compared  with  milling 137, 140 

flats  produced  by 144 

spiral  gears 118, 122, 137 

spiral  gears,  change  gears  for 123 

teeth  produced  by 143 

worm-gears 216,  231,  235 

worm-gears,  change  gears  for 216 

worm-gears,  suggestions  for  refinement  in 239 

Robbing  machines,  advantages  of  differential  mechanism 134 

calculating  change  gears  for 123,  216 

output  of 142 

Horsepower  transmitted,  by  herringbone  gears 87, 92 

by  worm  gearing 187, 188 

Hydraulic  turbines,  herringbone  gears  for « 97 

"  Jack-in-the-box"  mechanism 113, 1 20 

Lead,  of  spiral  gears 3»  5>  20 

of  worms *57 

Linear  pitch  of  worms *57 


LOAD  —  SHAPING  271 

PAGE 

Load,  allowable,  in  worm  gearing 172 

and  efficiency  of  worm  gearing 170 

safe,  on  worm-gear  teeth 174 

Lubricants  for  worm-gears .- : 187 

Machine  tools,  herringbone  gears  for 96 

Materials,  for  herringbone  gears 89 

for  worm  gearing 176 

Milling  and  hobbing  compared -. 137, 140 

Milling,  herringbone  gears 76 

spiral  gears,  angular  position  of  table 106 

spiral  teeth 101, 104,  no 

Milling  cutters,  feed  marks  produced  by 137, 153 

for  herringbone  gears 76 

for  spiral  gears 5,  20, 105, 107 

for  spiral  gears,  formula  for 25 

Number  of  flutes  in  worm  hobs,  general  formula  for 263 

Output  of  gear-cutting  machines 142 

Outside  diameter,  of  spiral  gears 6,  20 

of  worms 158, 166 

of  worm-gears 159, 167 

of  worm-gears,  table  for  calculating 167, 168 

Overheating  in  worm  gearing 184 

Pitch  diameters,  herringbone  gears 84, 85 

spiral  gears 4,  20 

worm 158, 166 

worm-gear 158, 166 

Pitch  line  velocity  and  efficiency  of  worm  gearing 172 

Pitch  of  worms , 157 

Planing,  herringbone  gears 77 

spiral  teeth 98, 116 

Power  transmission,  by  herringbone  gears 87, 92 

by  worm  gearing 187, 188 

requirements 71 

Pressure  angle  of  herringbone  gears 82, 85 

Reinecker  universal  gear-cutting  machine in 

Rolling  mills,  herringbone  gears  for 97 

Rotation,  direction  of,  in  spiral  gears 31 

Safe  load  on  worm-gear  teeth 174 

Self-locking  worm  gearing 176, 191 

Shaping  spiral  teeth 98 


272  SPIRAL  —  THREAD 

PAGE 

Spiral  angle  of  herringbone  gears 81, 85 

Spiral,  direction  of,  in  gears 31 

Spiral-fluted  hob  angles 248,  250 

Spiral  gears i 

angular  position  of  table  when  milling 106 

center  angle i 

cutters  for  milling  teeth 5,  20,  105,  107 

definitions I 

examples  of  calculations , 6, 19 

formulas  for 20,  29, 33 

graphical  solution  of 8 

hobbing '. 118, 122 

hobs  for 137 

lead 3,  5,  20 

methods  of  forming  the  teeth 98 

problems,  shafts  at  45-degree  angle 48 

problems,  shafts  at  any  angle 59 

problems,  shafts  at  right  angles 38 

problems,  shafts  parallel 33 

problems,  special  case 66 

procedure  in  calculating 31 

rules  and  formulas i,  4,  20 

tooth  angle 2 

tooth  dimensions 5, 6,  20 

Spiral  teeth,  milling ror,  104,  no 

planing  or  shaping 98, 116 

Spirals  on  hobbing  machines,  calculating  gears  for  cutting 123 

Spur  gears,  action  of 71 

Steam  turbines,  application  of  herringbone  gears  to 95 

Strength  of  herringbone  gears 86 

Strength  of  worm-gear  teeth 174 

Taper  hob  for  cutting  worm-gears 223 

Taper  hobbed  worm  gearing,  efficiency 226 

Teeth,  dimensions  for  spiral  gears 5, 6,  20 

dimensions  for  worm 158, 166 

hobbing  spiral  gear 118, 122, 137 

in  hobs,  shape  of 151 

in  spiral  gears,  methods  of  forming 98 

produced  by  gear  hobbing 143 

proportions  herringbone  gears 82,  85 

safe  load  on  worm-gear 174 

worm-gear,  forming 212 

Thickness  of  teeth  in  spiral  gears 6,  20 

Thread  angle  and  efficiency  in  worm  gearing 172 

Thread  dimensions  of  worm 158, 163, 166 

Thread  tool  for  worm  hob 245 


THREADING  —  WORM  273 

PAGE 

Threading  worms 227 

Throat  diameter  of  worm-gear 158, 166 

Thrust,  direction  of,  in  spiral  gears 31 

Tooth  angle  of  spiral  gears 2 

Turbines,  application  of  herringbone  gears  to  steam 95 

herringbone  gears  for  hydraulic 97 

Velocity,  of  herringbone  gears ... 89 

of  pitch  line  and  efficiency  in  worm  gearing ~] 172 

ratio,  in  worm  gearing 160 

Width  of  face  of  herringbone  gears 81 

Worms,  methods  of  forming  the  teeth  in 98 

minimum  length  of 160, 167,  251 

pitch  and  lead  of 157 

rules  for  dimensions  of 157, 166 

with  large  helix  angle 167 

Worm-gears,  abrasion 183 

applied  to  automobile  drives 180 

bearing  friction 196 

change  gears  for  bobbing 216 

cutting  of  Hindley 230 

dimensions 158, 166 

efficiency  of  taper  hobbed 226 

fly-tool  method  for  cutting. 219 

forming  the  teeth  of 212 

gashing 213,  231,  235 

hobbing 216,  231,  235 

hobs  for 244 

lubricants  for 187 

model  drawing  of 169 

self-locking 176, 191 

suggestions  for  refinement  in  hobbing 239 

table  for  calculating  outside  diameter 167, 168 

taper  hob  for  cutting 223 

teeth,  reducing  flats  on  hobbed 238 

teeth,  safe  load  on 174 

Worm  gearing 156 

applications 161 

center  distance 160, 166 

dimensions  of 156, 166 

efficiency 170, 179, 181,  226 

examples  of  calculations 162 

for  freight  elevator 177, 183, 184 

Hindley 202,  230 

horsepower  transmitted  by 187, 188 

load  and  efficiency 170, 174 


2  74  WORM  —  WUEST 

PAGE 

Worm  gearing,  materials  for 176 

objections  to  Hindley 209 

overheating 184 

points  in  design 175 

rules  and  formulas  for 156, 166 

self-locking 176, 191 

theoretical  efficiency 179 

velocity  and  efficiency 172 

velocity  ratio 160 

Worm  hobs,  flutes  in 246,  255,  263 

thread  tool  for 245 

Worm  teeth,  dimensions  for 158, 166 

Worm  thread  dimensions,  table ^3 

Worm  threading 227 

Wuest  herringbone  gears 69, 80 


14  DAY  USE 

RETURN  TO  DESK  FROM  WHICH  BORROWED 

LOAN  DEPT. 

This  book  is  due  on  the  last  date  stamped  below,  or 

on  the  date  to  which  renewed. 
Renewed  books  are  subject  to  immediate  recall. 

REC'D 


JUN  2  8  1980 


LD  21A-60m-4,'64 
(E4555slO)476B 


General  Library 

University  of  California 

Berkeley 


YC  6S87 


416063 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 


